How To Graph Equations - Linear, Quadratic, Cubic, Radical, & Rational Functions

today we're going to focus on graphing linear equations including those with inequalities quadratic equations we're going to talk about transformations graphing radical functions cubic functions absolute value equations rational expressions exponential and even logarithmic equations so let's begin let's start with the basics let's say if we want to graph y is equal to 2x plus 3 now there's many ways to graph it so the first way we're going to use is um we're going to make a table let's keep things simple so let's say if you have your XY table let's plot some points so we're going to plot the point negative 1 0 & 1 we don't really need that many points okay so at negative one if we plug in negative 1 for X so 2 times negative 1 plus 3 that's negative 2 plus 3 that's 1 if you plug in 0 you should get 3 if you plug in 1 2 plus 3 is 5 and then you just gotta plot those points so here's a negative 1 on the x-axis and over here Y is 1 that's our first point when x is 0 and Y is 3 we have this point here I'm going to plot another point because my graph doesn't go up to 5 so I'm going to plot negative 2 if we plug in negative 2 for X that's 2 times negative 2 that's negative 4 plus 3 we get negative 1 so that negative 2 is around somewhere over here and then you could just connect those points of line this is just a rough sketch my graph is not that perfect but that's the general idea of how to plot or how to graph a linear equation using the table you just got to pick a few points and just plug it in but now let's see if we can graph another question just like that but without I'm making a table let's say if we have y is equal to negative 2x plus 1 okay this equation is in slope-intercept form MX plus b b is 1 that's your y-intercept so that's where your graph is going to start at 1 M is the slope the slope is negative 2 which is the same as negative 2 over 1 this is your rise and this is the run the rise over run so every time you move one unit to the right you need to go down 2 units so wanting it to the right down to our next point is over here and if we go one unit to write down 2 here's our next point and once you need to write down 2 it's going to be somewhere over here and then we just got to connect those points with a line so that's the slope intercept method you just plot the y-intercept and then use the slope to find each successive going so now I'll try another example just like that but using fractions so let's say if you have this equation y is equal to negative 3/4 X plus 5 so one two three four five one two three four let's go up to eight five six seven eight so the y-intercept is 5 so that's going to be the first point now the slope is negative 3 over 4 so negative 3 is the rise and four is the run so we're going to travel 4 units to the right and we're going to go down three units so our next point is over here and then we're going to do the same thing we're going to travel 4 units to the right towards 8 and down 3 so we're going to be down at negative 1 and then just connect those points with the line so go ahead and try this example feel free to pause the video and give this problem a shot and when you're ready just unpause it okay so let's begin let's begin by plotting the y-intercept which is at negative 2 and the slope is 2 over 3 so our rise is 2 and the run is 3 we're going to go up 2 units and then over 3 units to the right and then same thing up 2 over 3 so we got these three points to plot and then once you feel like you have enough fluence you really only need at least two to make a good line once you have enough points you could just slot it so that's how you can graph linear equations in slope intercept form but now let's talk about how to graph it when it's in standard form standard form is ax plus B Y is equal to C so let's say if we have this equation 2x plus 3y is equal to 6 now what you want to do is you want to find the x and y intercepts so to find the x intercept plug in 0 for y so this becomes 0 so 2x equals 6 if you divide both sides by 2 you get x equals 3 so you get the point 3 comma 0 so that's the x-intercept to find the y-intercept plug in 0 for X so you get 3y is equal to 6 and if you divide both sides by 3 y is equal to 2 so your y-intercept is 0 comma 2 and that's what we want to do if you have an equation in standard form and then once you have those two intercepts just plot those two points and connect them with a line so 1 2 3 4 1 2 3 4 ok so the first point is that 0 2 the next one is at 3 0 and then just connect those two points with a line that's the simplest way to graph an equation in standard form so let's try another example let's say if we have 3x minus 2y is equal to 12 go ahead and try this and then plot it when you're ready so let's find the x-intercept if we plug in 0 for y the 2y disappears so we're just going to get 3x is equal to 12 and if we divide both sides by 4 12 divided by mu by 3 so I've divided by 3 is 4 so we get the point 4 comma 0 and for the y-intercept we're going to plug in 0 for X so we're just going to get negative 2y is equal to 12 and we're going to divide both sides by negative 2 so we're going to get Y is equal to negative 6 excuse me which is the point 0 negative 6 so we have 0 negative 6 and 4 0 and we're just going to connect those two points with a line I don't have a ruler so that's just my graph won't look perfectly straight but you get that the point though so now you know how to plot it in standard form how would you graph an equation that looks like this y is equal to 3 how would you plot that all you got to do if y equals the number it's just a horizontal line F 3 that's how you graph it and if you get a question looks like this let's say X is equal to 2 it's simply a vertical line at 2 so it looks like this so that's how you can graph a linear equation if Y or x equals a constant but now let's talk about graphing inequalities let's say if you have this question y is greater than 2x minus 4 and Y is less than or equal to negative 3 over 2x plus 3 by the way if it's in standard form make sure to convert it to slope-intercept form which is what we see here keep in mind slope-intercept form is like this y equals MX plus B you want Y to be on one side of the inequality this is going to make it a lot easier to graph it okay so go ahead and plot these equations in the mean time so let's plot the first one so the y-intercept is negative four and the slope is two so every time we move one unit to the right where you can go up to one unit two right up to the slope is two over one and notice that Y is greater than 2x minus 4 but not greater than or equal to if it's just greater than we need to use a dashed line now for this one this is going to be a solid line because it's less than or equal to negative 3 over 2 X plus 3 but for this one make sure you use like a dashed line so now let's graph the other equation so the y-intercept is at 3 and the slope is negative 3 over 2 that means as you move 2 units to the right the Run is to the rise is negative 3 so when you scale down 3 so over 2 down 3 and then over 2 down 3 again and then over 2 down 3 and it's going to be a solid line okay so now we need to know which region to shade there's four regions in the graph this is the first one this is the second one this is the third and this is the force ignore the X and y-axis look at the the four regions created by the green and yellow line when you shade it it's going to be one of those four regions so let's focus on this equation first y is greater than 2x minus 4 so the Green Line is the 2x minus 4 line greater than I'm going to shade in white is anywhere above that line so that entire region that's greater than the line then the green line now for the yellow line y is less than or equal to so it's below the yellow line I'm going to shade that in purple so this is the region below the yellow line now we need to shade the region where it's true for both and it's true in this region here that's where we have both the purple and the white line so this is where the answer is on a test this is the only region you should shade the other ones erase the marks that you made and that's how you choose the right region to shade so let's try another example let's say if X is greater than negative 3 y is less than or equal to 5 and Y is greater than X minus 3 so X is that negative 3 that's going to be a vertical line and it's going to be a dashed line because it's just greater than but not equal to and y is less than 5 so that's a horizontal line at 5 but it's a solid line and Y is greater than X minus 3 so the slope intercept I mean the intercept the y intercept is at negative 3 and there's a 1 in front of X so the slope is 1 so every time you move one to the right you go up 1 and it's going to be a dashed line now we need to know where to shade it so let's look at this first equation X is greater than negative 3 it's greater than negative 3 to the right anywhere to the right of this function now Y is less than 5 is less than 5 below this line anywhere below this line and Y is also greater than X minus 3 it's greater than X minus 3 above this line so we need to choose the region in which all arrows are located in if you notice it's true within this triangle so this is the region that we shade because all three equations are true in that region and that's how you do it ok so now let's move on to quadratic equations so let's say if you have this graph y is equal to x squared plus 2x plus 3 notice that it's in standard form ax squared plus BX plus C now for this example we're going to make a data table we're going to plot points so if you want to make a data table you want to find the vertex first to make your life a lot easier and the x-coordinate of the vertex is negative B over 2a that's where the axis of symmetry is so let's say if you have a graph that looks like this this equation will get this x-coordinate and this is useful because let's say if you plot this point and you choose this point they're going to have the same y-value and these two points will have the same y-value so if you want five points you really need to find three and two of them will be the same as the other two so a is 1 this is a and B is two so let's use the equation X is equal to negative B which is negative 2 over 2 times a or 2 times 1 so the vertex is that x equals negative 1 now we're going to pick five points but our center point is going to be the vertex negative 1 we're going to pick two points to the right of that point and two points to the left two points to the left of negative 1 is negative 2 and negative 3 and two points to the right of it is 0 and 1 so now let's plug in these numbers in let's start with negative 1 if we plug in negative 1 for X we get negative 1 squared plus 2 times negative 1 plus 3 so now it's 1 minus 2 plus 3 which is 2 if we plug in let's say 0 for X there's just going to be 0 squared plus 2 times 0 plus 3 which is just 3 so 0 and negative 2 are going to have the same y value because they're equally distant from the vertex let's plug in 1 it's easy to plug in 1 then negative 3 so 1 squared plus 2 times 1 plus 3 that's going to be 6 so negative 2 B also is going to have the same y value of 6 all right so let's make our graph the majority of the graph is at the top so 1 2 3 4 5 6 and most of our x-values on the left side so 1 2 3 4 1 2 3 4 so the first point is at 0 negative 1 2 which is here that's the vertex next we have the point 0 3 and a negative 2 3 and then we have 1 comma 6 which is up here somewhere and negative 3 comma 6 so now we can graph it and it's going to look like this table if I miss this point so that's a rough sketch the axis of symmetry is this line right here it's simply x equals negative 1 is the x-coordinate of the vertex so that's the AOS axis of symmetry if you need to find the minimum value the minimum value is the y coordinate of the vertex is right there that's where the minimum is located let's try another example like this for the sake of practice so let's say if we have y is equal to 2x squared minus 4x plus 1 so let's find the vertex first its negative B which is negative negative 4 over 2a a is 2 so this is a this is B so this is equal to 1 so let's make our table and we need two points to the left of one at 0 and negative one and two points to the right two and three so let's begin by plug in 1 for X so 2 times 1 squared minus 4 times 1 plus 1 so this is 2 minus 4 plus 1 and that's equal to negative 1 now it's easier if we plug in zero instead of two so two times zero squared minus 4 times zero plus one is one the next point we're going to plug in is a negative 1 this is also me 1 by the way so this is negative 1 squared is 1 times 2 negative 4 times negative 1 is 4 so we're going to get 7 for this point which means this one is also going to be 7 ok so now let's make our graph so it's going to go up to at least seven and most of our x-values will be on the right side so our vertex is that one negative one which is over here our next point is that zero one and we have another one at two comma 1 and then 3 comma 7 which is all the way here and negative 1 comma 7 which is right where I put the 7 so then we can graph it and here is the axis of symmetry it's x equals the x-coordinate of vertex so that's how you can plot quadratic equations and using the table but now let's say if we don't want to make the table let's say if you want to graph it a simpler way so let's say you have y is equal to x squared minus 2x plus 3 find the vertex as usual it's negative B over 2a so it's negative negative 2 over 2 times 1 so therefore X is 1 now you want to find the y-coordinate at that point so if you plug in 1 for watt for X you get 1 squared minus 2 times 1 plus 3 that's 4 minus 2 which is 2 so we got the point 1 comma 2 so our graph is initially right here so here's what you can do let me show you a pattern let's say if you plug in 1 for X you're going to get 1 for y right if you plug in 2 for X you're going to get 4 for y if you plug in 3 for X you're going to get 9 for y so once you have your vertex and this is 1x squared by the way because if what's 2 x squared you have to double everything to find the next point after you have the vertex move one to the right and then up one and do the same thing for the left side one to the left up one because one squared equals one now 2 squared is 4 so starting from your vertex as you move to the right go up 4 1 2 3 4 same thing here - to the right of 4 and that's a quick way you can plot this equation without needing data table so now we're going to focus on graphing these equations but in vertex form so let's talk about how to convert a quadratic equation from standard form to vertex form now there's something called completing the square so here's what you need to do you have x squared plus 6x we're going to focus on that number 6 take half of that number and then square it half of 6 6 3 and we're going to square it we still have the minus 3 now notice that we added 9 to the right side so we change the equation to balance that we need to add 9 to the left side or subtract 9 from the right side since I don't want to change the left side I'm just going to put minus 9 on the right side so if you add 9 and subtract 9 you haven't changed the value of the equation imagine if you have a bank account let's see have a thousand bucks in your bank account if you add a hundred and take away 100 it's still the same so the two equations are equal to each other but this can be factored if you factor it here's the shortcut method it's going to be what you see here X and whatever this sign is plus whatever this number is before you square it 3 squared it's always going to work out that way and then these two numbers you can combine so now you have it in vertex form which is X minus H squared plus K the vertex is H comma K so in this form you can easily see what the vertex is so in our example notice that H and negative H they have the opposite sign so you got to switch it from three to negative three this is H and K the sign is the same as both positive so whatever this number is that's K you don't change it so it's negative three negative twelve I'm going to go by twos two four six eight ten twelve so that's negative twelve two four six eight and so forth so we have the point negative three negative 12 which is somewhere in its region and since one squared equals one as you move one to the right go up one so that's going to be somewhere over here at negative two slash negative eleven one to the right up one and then as you go to to the right you need to go up four so from negative twelve is going to be at negative eight and then two to the right up four now if we go three to the right three squared is nine we need to go up nine so that will take us to negative 3 and 3 to the right here which will be at negative 6 with the Y value will also be at negative 3 if we go 4 to the right 4 squared is 16 so we'll be at the vertex is at negative 3 so 4 to the right 1 X will be 1 negative 3 plus 4 is 1 and negative 12 plus 16 is 4 so we'll be at 1 comma 4 which is somewhere over here and if we go forward to the left negative 3 minus 4 that's negative 7 and the Y value should still be the same it should be around 4 as well and now we can graph it so that's a rough sketch all my sketches are rough so that's what you can do if you have an in vertex form let's try another example so let's say if you have y is equal to negative X minus 3 squared plus 4 now this negative sign tells you that the graph it's going to open downward so our vertex change negative 3 to positive 3 but don't change the 4 that's our vertex 3 comma 4 so let's plot that point so here's 3 comma 4 and we're going to need more points on the right side now we know the graph is going down so as we move we run to the right it's going to go down 1 and once the left down 1/2 to the right it's going to go down 4 from starting from the vertex so it's going to be at 0 on the x-axis 2 to the left is also going to be done for now if we go 3 to the right from the vertex which will be over here we need to go down 9 so we should be at negative 5 which is over here and if we go three to the right we'll be at the the y-axis and we'll be down five as well so this graph looks something like this and that's how you graph it now what about this one let's say there's a 2 in front of the parentheses so how does it modify our process so the vertex we know it's negative 1 you change the plus 1 to minus 1 comma negative 2 this is going to say K is going to be the same my lines are never straight so notice that it's positive - so that means the graph is going to open upward so let's plot the vertex so it's at negative 1 comma negative 2 which is over here somewhere now we won't be moving one to the right and up 1 because our parent function is y equals 2x squared there's a 2 in front so if you plug in 1 for X you're going to get 2 for y if you plug in 2 for X you get a fly so as we move one to the right so I have to go up one it's going to go up 2 because it's going to be times 2 so one to the right up to that should be over here somewhere one to the left up to start from the vertex as we go 2 to the right we need to go up 8 instead of 4 2 squared is 4 but we got to multiply by 2 so if we go up 8 we should be at 6 from negative 2 and 2 to the left when you scale up 8 as well it's going to be at the same point and so that's how you graph it you and the axis of symmetry if you have to find it it's x equals this value negative one so here's the line of symmetry it's symmetric about that line and this graph has a minimum whenever you have a positive number here the graph always has a minimum and the minimum value is simply the y coordinate of the vertex is negative 2 when it opens downward let's say if you have negative x squared then you have a maximum 2 sometimes you may find make it like word problems where they ask you what is the maximum value of this function just find the vertex and look for the Y value so that's it for quadratic equations now let's move on to absolute value functions the absolute value of X looks like this it's a v-shape and it has a slope of 1 so let's say if we want to graph the absolute value of X minus 3 plus 4 first need to find the vertex so we need to move 3 to the right because we have a minus 3 on the inside and up 4 kind of like the last problem the vertex is 3 comma 4 you change what's on the inside but you don't change what's on the outside now the slope is 1 for this problem so as you move one to the right you need to go up 1 1 to the left of 1 it's symmetrical just like the quadratic function but notice that it's y equals x rather than y equals x squared so when X is 1 Y is 1 when X is 2 y is 2 so the slope is going to be 1 and then just keep moving one to the right and up one and then just plot the graph so that's how you can graph absolute value functions but now let's say if we have y is equal to negative 2x plus 1 minus 3 now the fact that it's negative on the outside means that it's going to open downward instead of upward and the fact that we see a 2 the slope is now 2 instead of 1 so as we move one to the right is going to go down 2 and it's going to the slope is constant it's going to always be 2 so first let's plot the vertex which is negative one negative three so it's over here somewhere negative 1 negative 3 and we said that's going to go downward so as we move one to the right it's going to go down to let me use a different color and one to the left and down to and one more time one to the right down to run to the left down to so it's going to look something like this so that's how you can graph absolute value functions so let's move on to cubic functions let's say if you have Y is equal to X cubed the parent function for this graph looks something like this that's just sort of rough sketch so let's say if you want to graph this equation the center point or the worthy where it should be the origin it's at a 1 comma 2 you change the negative one but not the two so the shift in moves one to the right and it moves up to what you see on the inside is the horizontal shift which is one to the right this is the vertical shift it's up to so the origin is now at 1 comma 2 now we can draw a rough sketch if you want we can do something like if we want to but let's actually get point to make this graph for more accurate so our parent function is y equals x cubed 1 to the third is 1 2 to the third is 8 so to get the first two points as you move one to the right um go up one one to the left go down one and now if you move to to the right you need to go up eight and we really don't have the space for that but it's probably going to be somewhere over here and if we move to the left we need to go down 8 now this we can plot because that's somewhere that's going to get negative six so down eight will be all the way over here so now we can get a more accurate sketch so it's going to look something like that so let's try one more example with cubic functions so let's say if you have y equals instead of putting negative two I'm going to put negative 1/2 because I don't want to go up 16 let's say if we have this and it's minus 1 we know the center is going to be at negative 3 negative 1 so it's going to shift three units to the left and down one so it's at negative three negative one which is in this area and let's say if we move one unit to the right well by the way notice that it's negative positive X cube looks like this negative x cube looks like that so we have a decrease in function rather than an increase in function so as we move one to the right one this is supposed to be cubed by the way 1 to the third is one but we got a half so we should move down a half so one to the right I'm just going to go down a half because of this number one to the left is going to go up a half now 2 to the third is 8 what times 1/2 is 4 so as we move 2 to the right it's going to go down 4 so from negative 1 to negative 5 and as we move to to the left this point should be actually here let me fix that once to the left should have been up 1/2 2 to the right that's down 4 so it's over here at negative 5 and 2 to the left we need to go up 5 I mean up 4 so 1 2 3 4 it should be over here somewhere so now let's see if we can make our graph so we have to be the function has to be decreased instead of increasing only because of the negative sign so that's a rough sketch for that graph so now let's go over radical functions let's say if we have this function of square root of x the parent function for that looks like this and if you have let's say negative square root X the negative in front causes it to flip over the x-axis or it's a reflection across the x-axis so it looks like this and let's say if you have a negative on the inside it reflects across the y-axis so it looks like that and if you have a negative both on the inside and on the outside it reflects across the origin so it looks like this the way I like to see this let's say if here X is positive and Y is positive so x and y are positive in quadrant one so it goes towards quadrant one here X is positive but Y is negative so that's quadrant four X is positive as you go to the right but Y is negative as you go down here X is negative Y is positive that's quadrant two and when x and y are both negative its quadrant three that's a an easy technique you can use that to see where the graph is going to go for radical X but now let's apply it in a problem so let's start with a straightforward one let's say if we have y is equal to the square root of x plus three minus two so the origin it's shifted 3 to the left and down to so it's going to start over here now the parent function is the square root of x the square root of 1 is 1 and the square root of 4 is 2 and the square root of 9 is 3 so as we move one to the right go up one as you move for to the right go up to so that's going to be somewhere over here and as you move nine to the right from your original point and go up three so right now we're at negative three if we move nine to the right we should be at positive 6 and going up three we need to be at y equals one so and that's our graph it starts from here now one thing I should have mentioned is domain and range what is the domain for this function and what's the range the domain is the X values the ranges the Y values if you want to find the domain analyze it from left to right so it starts at the lowest x value is that negative three there's nothing before that and the highest x value it goes to infinity so your domain is from negative 3 to infinity and it includes negative 3 the range is the Y values the lowest Y value is that negative 2 which is that number and this arrow it keeps going up but it goes up slowly and slowly but it goes up to infinity so the range is from negative 2 to infinity now just to review for linear equations like let's say Y is equal to 2x plus 3 I'm just going to draw a rough sketch so it starts at 3 for slope of 2 let's say it looks like this the domain for a linear function is just negative infinity to infinity the range is the same thing now the next one that we considered was a quadratic function let's say if you have 3 plus x squared the graph is going to start after me and it's going to open upward it's been shifted three units up so the domain for any quadratic function the X could be anything it's always going to be negative infinity to infinity however the range will change notice that the lowest Y value is three but it goes up to infinity so the range is from three to infinity now if you have a downward parabola let's say like Y is equal to five minus x squared your graph is going to start at five and it's going to open downward it's been shifted up five units so your domain for any quadratic function where you have x squared as Y or your highest degree it's always going to be negative infinity to infinity the range though is from let's see your lowest Y value is negative infinity because these arrows will go all the way down your highest y-value is at five so from low to high negative infinity to 5 it stops at 5 so for your domain you looking at it from left to right from the range you're looking at it from low to high or bottom to top now the next thing that we considered was an absolute value function so let's say if you have 2x minus 3 plus 1 so the graph shifts 3 units to the right up and when it starts here and the slope is 2 so I'm just going to draw a rough sketch your domain for any absolute value functions is negative infinity to infinity unless you have like a piecewise function your range notice that your lowest Y value is 1 but it goes up to infinity so it's from 1 to infinity and let's say if it opens downward let's say you have 3 minus absolute value of x plus 2 so it shifts two units to the left and it's shifted up three units so it starts here but there's a negative in front of the absolute value so it goes like this it goes downward with a slope of 1 so your domain is everything for this function as well but the range the highest value is 3 the lowest value is negative infinity search that's from low to high negative infinity and the stop sign 3 now for cubic functions let's say if we have X minus 1 cubed plus 2 as you move one to the right it's going to be shifted up 2 and there's no number in front of in front of here so it's just a 1 and because it's positive it's going to be increasing rather than decreasing so it's going to look something like this for this type of function your domain is also everything there's no restrictions on X your range is also everything as well it goes down to negative infinity and up to infinity so for any cubic function domain and range is just our row numbers so now let's go back to radical functions let's look at our next example let's safely have the square root of 5 minus 2 minus X so the first thing when a plot is where the first point so if you set the inside equal to zero to minus x is equal to zero you're going to get X is equal to two so it shifts two units to the right and it shifts up five units so it starts somewhere here the question is what direction is going to go is there going to go towards quadrant one towards quadrant two towards quadrant three or towards quadrant four because knowing the direction is important because we need that if we don't want to make a table so notice that we have a negative in front of the radical and a negative in front of X do you see that so X is negative and Y is negative x and y are negative and what quadrant has to be quadrant three so it's going to be going in this general direction so now we don't have a number in front of the radical so we can use our general transformation the square root of one is one the square root of 4 is 2 and the square root of nine is three so as we go one unit to the left it's going to go down one and as we this is X this is y so as we travel four units to the left it's going to go down to so it's going to be over here and as we travel nine units to the left starting from this point it's going to go down three so it's going to be over here now for this if you prefer you can always make a table so I'm going to do one example by making the table so you can see which points to choose and there is the graph right there so let's say if you have y is equal to the square root of x plus 1 minus 3 and you want to make a table the first point you want to plug in is negative one and it's going to be negative 3 that's the because that's the ship it'll shift someone to the left and down 3 now from that point let me just put this number here you want to go one to the right so let's plug in 0 if you plug in 0 for X the square root of 1 is 1 and 1 minus 3 is negative 2 now your next point you don't want to plug in 1 2 3 & 4 you want to plug in numbers that are perfect squares because you can square root it the reason why I plug in 0 is because I can square root 1 the next number I'm going to choose this 3 because 3 plus 1 is 4 and the square root of 4 is 2 2 minus 3 is a negative 1 now the next point I'm going to plug in is 8 because 8 plus 1 is 9 the square root of 9 is 3 and 3 minus 3 is 0 if I want another point I'm going to plug in 15 because 15 plus 1 is a perfect square of 16 the square root of 16 is 4 4 minus 3 is 1 so you want to plug in numbers that are perfect squares where the radical would simplify to an integer and that's going to make a life easier and basically the way I was plotting points for these other functions is based on this pattern of numbers so let me show you so we know it shifts one to the left down three so this point is this one negative 1 negative 3 and 4 radical X radical 1 is 1 so as we move one to the right it's going to go up 1 notice that we have the point 0 negative 2 and the square root of 4 is 2 so start from this point if we move forth to the right which will put us out to be it's going to go up 2 so notice we have this point 3 negative 1 and then the square root of 9 is 3 so as we go 3 to the right so one two three four five six seven eight nine we're going to go up to B so it's going to be over here which is a comma zero so notice this technique gives you the points without actually making the table and if you can master this technique you can graph equations at a very fast rate so our graph looks like this now keep in mind let's say if we put a 2 in front everything is going to double so instead of going as you go months to the right you're going to go up 2 instead of up one as you go four to the right you're going to go instead of up - you need to go up four which is somewhere over here and as you go nine to the right instead of going up three you need to go up six which will be somewhere over here so your graph will look something like this if it was y equals to square root x + 1 - 3 so the amount that you increase in your Y values just double it that's it if there's a 2 in front if there's a 3 just triple it so like if we use this pattern here let's save those will - it would be - square root of 1 which is 2 if we plug in a 4 it's going to be 4 if we plug in nine it's going to be six square root of nine is three times two is six so these numbers the amount that we increase the Y values will simply double okay so now let's move on to a different type of radical a radical with an odd root instead of an even root the square root of x what we considered was this one there's a invisible two that you don't see so if it's even it looks like this but when it's odd it looks like this instead so negative cube root of x is going to look like this by the way keep in mind this is the same as cube root of negative x so even if you reflect it across the x-axis or across the y-axis it's going to be the same because this graph is symmetrical about the origin so there's only two ways you can graph cube root of x this way if it's positive or this way if there's at least one negative if there's two negatives they can cancel and it's going to be the same as this one if you have a negative on the inside and on the outside but now that we know what the parent function looks like let's try some problems so let's say if you want to graph the cube root of x plus 1 plus 2 so first you want to find out where the graph begins so it shifts one to the left and up two so it's over here now let's look at our pattern of numbers that we're going to use to plot this equation so the cube root of 1 is 1 so as we move one to the right it's going to go up 1 and the cube root of 8 is 2 so as we move 8 to the right it's going to go up 2 so one to the right up one one to the left download because you know the general shape is like this it's increasing now as we go 8 to the right so starting from negative 1 we're going to stop at 7 it's going to be up to so it's going to be over here somewhere and as we go 8 to the left it's going to take us to negative 9 so it's going to go down 2 so it's going to be over here somewhere and so now we can just plot it so it looks something like that so that's how you can graph cube root functions now let's say if you have 1 over X the parent function for one of X looks like this and if you have negative 1 over X it looks like this it reflects over the Y or the x-axis doesn't matter be the same so let's try some problems let's say if we wish to graph 1 over X minus 2 so we have the general shape but you want to plot the vertical asymptotes the vertical asymptote is when the denominator is equal to zero so if you set the bottom equal to 0 you get x equals 2 that's the VA the vertical asymptote now there's also a horizontal asymptote which is y equals zero which is over here whenever the degree of the denominator is greater than the degree of the numerator the horizontal asymptote is at y equals zero now if you want to get points for this graph there's really only one point you need but let's make a table I would plug in 3 for X because 3 minus 2 is 1 1 over 1 is 1 so after me it should be at 1 the next point I would plug in would be something to the left of 2 let's say like 1 if you plug in 1 you're going to get negative 1 so it should be somewhere over here so then our graph it starts from the asymptotes and then it connects to this point and then it follows the horizontal asymptote so it looks something like that you can add more points if you want in this graph looks like that so that's how to graph 1 over X minus 2 so let's say if we have y is equal to 1 over X plus 3 plus 4 the vertical asymptote is at x equals negative 3 based on in this term here if we set equal to 0 now the horizontal asymptotes for this fraction is 0 because it's bottom-heavy the degree of dignity denominator is higher than that of the numerator but it's 0 Plus this number 4 so it's actually y equals 4 it's been shifted up 4 units so let's make our graph so the vertical asymptote is at negative 3 and a horizontal asymptote at 4 and let's plug in some points so I'm going to plug in one point to the right of negative three so that's going to be negative two and one point to the left of negative 3 that's negative 4 if I plug in negative 2 negative 2 plus 3 is 1 1 over 1 is 1 if I plug in negative 4 negative 4 plus 3 is negative 1 and so I got those points so negative 4 negative 1 it's around here and negative 2 you know I forgot to add 4 to each number so the Y value should be 3 and the Y value here should be 5 can't forget this 4 but I plotted at the right point for some reason and negative 2 it should be at 5 so somewhere over here now we can make our graph so this one's going to look something like this and this graph is going to look something like that as you can see it's been shifted left 3 up 4 now let's say if we have a negative value let's say Y is equal to negative 1 over X minus 1 plus 2 so the vertical asymptote if you set x -1 equal to 0 X equal to 1 the horizontal asymptote is going to be y equals 2 based on this number so here's the horizontal asymptote and the vertical asymptote is that one now notice that we have a negative sign so instead of the graph looking like this it's going to be like this instead so let's plug in points let's make a let's make a table so I'm going to plug in 2 for X and 0 for X 2 minus 1 is 1 but times negative 1 divided by 1 is negative 1 plus 2 so that's 1 if we plug in 0 it's going to be negative 1 over negative 1 plus 2 and we're going to get 3 if we plug in 0 for X so we have the point 2 comma 1 which is over here and 0 comma 3 which is here and so this graph is going to look like this and this one's going to look like that so now you know how to graph parent functions one of X functions now let's say if you have 1 over x squared it's very similar to 1 over X but both curves are above the x-axis if you have negative 1 over x squared it's going to be below the x axis it's going to look like this so this graph is symmetrical about the y-axis the other one 1 over x was symmetric around the origin so let's try one example let's say if we have negative 1 over X minus 2 squared plus 3 the vertical asymptote you set the bottom equal to 0 you're going to get x equals 2 the horizontal asymptote it's simply y equals 3 so for these types of rational functions always plot the asymptotes first so let's make a table I'm going to plug in 1 for X and 3 1 units the left and 1 unit to the right so if I plug in 3 3 minus 2 is 1 1 squared is 1 negative 1 over 1 is negative 1 plus 3 that's 2 if I plug in 1 1 minus 2 is negative 1 but if you square it it becomes positive 1 negative 1 over 1 is negative 1 plus 3 is 2 notice that they have the same y-value which means they're symmetric around about the the y axis but in this case it's going to be symmetric about the vertical asymptote because it's been shifted two units to the right so if we plot 1/2 that's over here I mean that's 2 recompensed room that 1/2 1/2 is over here but we can see the symmetry around the vertical asymptote so it looks like that so let's go over some different types of rational functions let's say if you have x squared plus 5x plus 6 divided by x squared plus X minus 2 now this function might seem complicated but it's not really we've pretty much covered this material what you should do is you should factor everything so to factor x squared plus 5x plus 6 think of two numbers that multiply to 6 but add to the middle term 5 2 times 3 is 6 2 plus 3 is 5 so it's going to be X plus 2 times X plus 3 so factor on this one look for two numbers that multiply to negative 2 would add to 1 so that's positive 2 and negative 1 2 plus negative 1 is 1 so it's X plus 2 X minus 1 notice that the x mark plus 2 is canceled so if you set X plus 2 equal to 0 x equals negative 2 is a hole the vertical asymptote is based on this one because the X minus 1 couldn't be canceled so x equals 1 is the vertical asymptote now you might be wondering what is the horizontal asymptote now notice that the degree of the denominator is 2 and the degree of the numerator is 2 as well when it was bottom-heavy when a degree of the denominator was higher than that of the numerator like 1 over X or 1 over x squared the horizontal asymptote was y equals 0 plus if there was a constant here it was plus 3 then add 3 but we don't have that 3 there so whenever the degree is the same divide the coefficient so 1 divided by 1 is 1 so the horizontal asymptote is y equals 1 for this problem so now let's graph it so first let's plot the vertical asymptote at one next let's plot the horizontal asymptote at one so now at this point we know the graph is going to look like 1 over X it's going to be very similar to it so we just need a few points one point I would plug in is the hole negative 2 because that's important if we plug in negative 2 we just need to plug it into the surviving equation so it's a negative 2 plus 3 negative 2 minus 1 negative 2 plus 3 is 1 and negative 2 minus 3 is negative 3 so we get the point negative 1/3 I would also plug in one number to the right of the vertical asymptote as we did before like 2 let's see if I can fit it here 2 plus 3 is 5 and 2 minus 1 is 1 so we just get 5 and plug in 1 number to the left of it so let's say 0 0 plus 3 is 3 and 0 minus 1 is negative 1 so we get negative 3 so negative 2 comma negative 1/3 is somewhere over here and 2 comma 5 is up here somewhere and 0 negative 3 is down here so we can see how this graph is going to look like it starts from 1 Assam so it connects to these points and it follows the other one however we did say this is a hole so this should be an open circle I know drew a closed circle but ignore the yellow point it should be a hole there now for the other one we know what the general shape is going to be it's going to look something like this you can add more points if you want but that's a rough sketch of the graph so let's try another function let's say if we have y equals x squared plus let's make that out 8x squared divided by 4 minus 2 x squared plus 1 so let's take out a 2 from the bottom and notice that it's going to be two minus x squared now we're going to set that part equal to zero and we're going to solve for X so X is equal to plus or minus radical two so we have two vertical asymptotes in this problem now to find the horizontal asymptote notice that the degree of the numerator which is x squared is the same as that of digital via so what we're going to do is divide the coefficients eight divided by negative two so the horizontal asymptote is at negative four but we have a constant in front so shifted up one so negative four plus one so it's at negative three so now we have the vertical asymptote and the horizontal asymptote the square root of 2 is about 1.4 so the vertical asymptote is somewhere between 1 and 2 and we have another one at negative 1 point 4 and the horizontal asymptote it's at negative 3 now we know how the graph is going to look like on the right side on the left side it can be like this or it can be like that it can be like this or it can be like this in the middle a lot of things could happen you can get something like this you may get a curve that looks like this or sometimes they can even cross the horizontal asymptote so for the middle we got a plug in point to find out what's going to happen so let's make our table for this one I would definitely use a table let's plug in one point to the left of radical two so let's plug in positive two for X so that's going to be six well 2 squared is 4 8 times 4 is 32 and let's see on the bottom we have 4 minus 2 times 2 squared so that's like 4 minus 8 which is negative 4 and so we should get negative 8 so 2 comma negative 8 so it's like all the way over here all we need is one point in this region so we know the graph looks something like this in this area so let's plug in another point let's say like a negative 2 and notice that it's squared so regardless if we plug in 2 or negative 2 the Y values can be the same so because it's both squared is going to be symmetric or symmetrical so a negative 2 is going to be like negative 8 again so this is going to be the same so the fact that it's symmetrical means that we're probably going to have something like this or something like this so let's plug in zero first if we plug in zero for X everything is zero so it's over here let's say if we plug in 1 and one a negative one is going to be the same if we plug in one it's going to be 8 over 4 minus 2 so that's 8 over 2 which is 4 so that's like over here somewhere so we can see what's happening this graph is going to look something like this at the center so between two vertical asymptotes you can't be certain what's going to happen but if you see that both of these terms are even like it has an even exponent you know that is going to be some sort of symmetry it can be like this or it can be like that if it was odd it could be like this at the center between the two vertical asymptotes or it could be like that you just got a plug in points to find out for sure so that's it for that particular function now this one more thing we got to mention and that's rational functions involving a slant or an oblique asymptote so notice that's a degree of the numerator which is 2 is greater than that than the degree of the denominator which is 1 when the numerator has when a numerator is one degree higher than the denominator you're going to have a slight asymptote now what we should do first is we should factor everything before we begin if we multiply these two numbers two times five is ten and two numbers that multiply to positive ten but add to negative three doesn't happen it doesn't work because two times five will add up to seven negative two and negative five will add up to negative seven so we can't factor the numerator if we could we should we know the vertical asymptote is going to be x equals two now whenever the numerator is higher than the denominator there is no horizontal asymptotes but we do have a slant so we need to do long division to find a slant asymptote but we don't have to complete the entire long division process there's a certain point where we can stop where it would suffice and we can have the answer two x squared divided by X is two x after you divide multiply two x times X is 2x squared + 2 x times negative 2 is negative 4x so let's subtract so here we get 0 negative 3x minus negative 4x which is like plus 4x you get X and bring down to 5x divided by X is 1 once you get this constant here that one you can stop so the slant asymptote is 2x plus 1 so now we can graph it so at 2:00 we have the vertical asymptote and to plot the slant asymptote notice that we have a linear equation in slope intercept form which we covet at the beginning of this video so we're going to plot the y-intercept which is 1 and the slope is 2 because it's a 2 in front of X excuse me so that's move one to the right go up 2 and as you move one to the right go up 2 as well and then we're going to plot it with a dashed line now these two asymptotes separate the graph into 4 regions so we can have a graph that looks like this it can look like that it can be over here or it could be here it's going to be in 2 out of those 4 regions and the only way to find out which region it's going to be is to plug in points we only really need to plug in two points one point to the right of the vertical asymptote and one point to the left so let's say if we plug in 3 for X 2 times 3 squared 3 squared is 9 times 2 is 18 negative 3 times 3 is negative 9 plus 5 so that's 9 plus 5 that's 14 on top 3 minus 2 is 1 so we get 14 so at 3 it's clearly above the it's way above the slant asymptote it's all the way up 14 which our graph doesn't go that high so we're going to say it's in this region somewhere we don't have enough space to plot it usually if it's in that region then chances are it's going to be in this region but let's find out for sure so let's plug in number less than 2 let's plug in 1 so 2 times 1 squared is 2 minus 3 times 1 which is 3 plus 5 over 1 minus 2 so that's negative 1 plus 5 that's 4 over negative 1 so it's negative 4 so at 1 it's at negative 4 so it is in this region let's go ahead and plug in zero two because zero is an easy number to plug in it's going to be five over negative two which is like negative two point five just to make this graph a little bit more accurate so our graph is going to look something like this now the more points you add the more accurate your sketch will be but it's up to you how many points you wish to add but usually on on a test you might get a multiple-choice question and all you need is one point to the left and one points to the right of the vertical asymptote and you can limonade the wrong answers to get the right answer or you could just use a graphing calculator so now we're going to focus on graphing exponential equations so let's say if we have y is equal to 2x plus 3 minus 1 now what you want to do is set the exponent equals two numbers zero and one so if you solve for X you get negative three and negative two these are the points that you should plug in into your X&Y table which I would recommend for this type of problem so if you plug in negative 3 for X negative 3 plus 3 is 0 2 raise to 0 is 1 anything raised to the 0 power is 1 and then 1 minus 1 is 0 if you plug in negative 2 for X negative 2 plus 3 is 1 and 2 to the 1 is 2 2 minus 1 is 1 now this number that you see here this is the horizontal asymptote before an exponential function so we're going to plot that so we have it at negative 1 and there's no vertical asymptotes for exponential functions your next thing after you plot the horizontal asymptote plot your two points negative 3 0 and negative 2 1 so the graph it starts from the horizontal asymptote and it follows those two points and that's how you can graph it you can add more points if you want but I think that sketch is good enough so let's try another one so let's say if we have Y is equal to 2 minus 3 raised to the X plus 1 so set the X for X plus 1 equal to 0 and 1 and solve for X so X is equal to negative 1 and X is equal to 0 and then make your table so we're going to plug in negative 1 and 0 for X if we plug in negative 1 negative 1 plus 1 is 0 and 3 to the 0 is 1 2 minus 1 is 1 if we plug in 0 0 plus 1 is 1 2 e to the 0 I mean 3 to 1 is 3 so we have 2 minus 3 which is negative 1 this number is the horizontal asymptote is the number is that it's the constant it doesn't have an X so the horizontal asymptote this time is that two and if we plot our points negative one comma one and zero negative one so start from the horizontal asymptote and connect the two points so that's how you can graph that particular exponential equation so the last thing we're going to look at is graph in logarithmic equations so let's say if you have log base 2 X plus 1 minus 3 so accidential functions are like they're like the opposite of logarithmic functions exponential functions have horizontal asymptotes logarithmic functions have vertical asymptotes now what you want to do is you want to set the inside part of the log function equal to 3 things rather than two things set it equal to zero which will give you the vertical asymptote always set it equal to one and set it equal to whatever this number is the base so the first one we get X is equal to negative 1 that's the vertical asymptote the next point we get zero and here we get one so let's make our table with these two points 0 & 1 if you plug in 0 for X you get 0 plus 1 log of 1 regardless of what the base is is always 0 so you have 0 minus 3 it's negative 3 if we plug in 1 1 plus 1 is 2 so we have log base 2 of 2 the 2's cancel and give you 1 and 1 minus 3 is negative 2 so that's why I chose those numbers because you get a nice whole number of integers to deal with which makes the easier to graph so the first thing you want to do is unplug asymptotes which is that negative 1 and then plug in your points so we have our first point at zero negative three which is over here and then one negative two so start from the vertical asymptote and then follow the two points and that's how you graph it let's try our final example so let's say Y is equal to two minus log base three X minus one so set the inside equal to zero one and the base in this case which is three so the vertical asymptote is X is equal to one and our other points are two and four so let's make our table so if we plug in 2 for X it's 2 minus 1 which is 1 log of 1 is 0 so 2 minus 0 is 2 if we plug in 4 4 minus 1 is 3 log base 3 of 3 is 1 and so 2 minus 1 is 1 so our vertical asymptote is that one and we have the point 2 comma 2 which is over here and the next one is 4 comma 1 so the graph has to start this way if it's going to connect to those two points and that's how you plot it so that's it for this video now you know how to graph almost any equation there's other equations too but for the most part we covered the most common equations that you'll see throughout your algebra trigonometry precalculus and calculus course so if you've seen any new equations that you have to graph you can apply the principles that you've learned in this video to those equations as well so that's it and thanks for watching this video and have a great day

Channel: The Organic Chemistry Tutor

Views: 883,468

Rating: 4.8812184 out of 5

Keywords: how to graph linear equations in slope intercept form, how to graph linear equations in standard form, how to graph quadratic functions in standard form, how to graph quadratic functions in vertex form, how to graph linear equations with inequalities, cubic functions, radical functions, how to graph equations, square root, how to graph rational functions, transformations, horizontal asymptote, vertical asymptote, slant asymptote, horizontal vertical and slant asymptote

Id: tfF_-Db8iSA

Length: 85min 59sec (5159 seconds)

Published: Tue Feb 16 2016