Reflections Over the X-Axis and Y-Axis Explained!

did you ever notice that the label on the hood of an ambulance is written backwards this reverse writing of the word ambulance is not a mistake when viewed through the rearview mirror of a car a mirror image correctly shows the word ambulance and the driver can quickly identify whether or not he or she should get out of the way in geometry we refer to a reflection as a mirror image [Music] unlike dilations a reflection is not a change in size a reflection is also not a change in shape for this lesson we will need to be familiar with common lines of reflection we know the x-axis as the horizontal line that passes through the origin the y-axis as the vertical line that passes through the origin the vertical line equation x equals K in this case k equals 3 the horizontal line equation y equals K in this case y equals negative 2 the line y equals x and the line y equals negative x [Music] in reflecting a point line or figure about a given line of symmetry we use the lowercase R notation this notation will specify the line of symmetry for which our reflection will occur for our first example we will be reflecting a point notice that our line of symmetry is the x-axis this is the line that we are going to be reflecting our point P with coordinates five four across to find the coordinates of the image P Prime we will count how many units the point P is away from the line of symmetry in this case four units X we will count the same number of units on the other side of the line of symmetry so again four units we have the location of the point P Prime with coordinates 5 negative 4 when reflecting over the x-axis we should notice that the value of the x-coordinate did not change however the value of the y-coordinate was the gated going from 4 to negative 4 this reflection rule applies to reflecting any point over the x-axis and where the coordinates XY will become x negative Y meaning that the value of the y-coordinate is negated now we can take a look at taking that same point P and this time reflecting it over the y-axis our approach here is still the same we're going to count how far away the point is from the line of symmetry in this case five units then we can count the same amount of units on the other side of the line of reflection and plot the point P Prime the coordinates of this image at negative 5 positive 4 notice that one reflecting over the y axis the x coordinate value is negated and the y coordinate value is unchanged favor reflection rule you can say that any point reflected over the y-axis with coordinates XY becomes negative XY where the x value is negated for our next example we are going to reflect a line segment our line of symmetry x equals negative 2 is a vertical line equation let's start by writing down the coordinates of the endpoints QT to find the coordinates of the image Q prime T prime will use the same strategy as the last example by counting how many units each endpoint is from the line of symmetry we see that the point Q is 2 units away from the line of symmetry x equals negative 2 and the point T is 6 units away we can go ahead and repeat those distances on the other side of the line of symmetry and plot the new points of Q Prime and T Prime now we can construct the image of the reflected line segment q Prime has coordinates of negative 4 7 and T Prime has coordinates of negative 8 3 now we can go ahead and try to reflect line-segment cutie over the line of symmetry y equals three a horizontal line equation notice here at the point t is directly on the line of symmetry just like pressing your finger up to the surface of a mirror and having it up here that another hand is directly touching it having a point directly on the line of symmetry will have it stay on the line of symmetry so T prime will have the same coordinates as T at 4 3 now the other end point Q still needs to be reflected again we're just going to count how many units Q is from the line of symmetry we count four units and we repeat that distance on the other side again another four units and plot the image of Q Prime finally we can construct line segment Q prime T Prime and identify the coordinates of Q Prime at 0 negative 1 and finally for our last example we are going to take a shot at reflecting a figure this case we will be reflecting triangle BCD across the line y equals x we can start by writing down the coordinates of the vertices of triangle BCD we could start by counting on a horizontal path how many units each point is away from the line of symmetry we see that B is 6 units away D is 7 units away and C is 12 units away now since y equals x is a vertical line equation when we count on the other side of the line of symmetry we are going to move vertically so we started horizontally on one side and now we're going to count vertically on the other side by matching the same number of units from each corresponding point we can find the coordinates of B prime C Prime and D Prime finally we can construct the reflected figure the coordinates of B Prime at 1/7 the coordinates of C Prime at negative 6 positive 6 and the coordinates of D Prime at negative 5 2 now we can compare the pre image and the image after the reflection by turning the graph and seeing just how these two figures are reflected over the line y equals x [Music] notice the coordinates of B at 7 1 and B prime at 1 7 C at 6 negative 6 and C Prime at negative 6 6 and D at 2 negative 5 and D Prime at negative 5 2 this relationship applies to any reflection over the line y equals x when you have a point P with coordinates X Y the X and y coordinate values are switched the y coordinate value comes first and the x coordinate value comes second the signs do not change and now we could take a look at reflecting this figure over the line y equals negative x again let's start by constructing the line of symmetry and writing down the coordinates of the vertices of triangle BCD notice here that the figure is resting on top of the line of symmetry that is in two different regions now we know that since Point C is directly on the line of symmetry C Prime will have the same coordinates of 6 negative 6 next point B is located above the line of symmetry we'll start by counting the number of the units horizontally it is away from the line of symmetry eight units the next step is to repeat that same number of units vertically on the other side of the line of symmetry to find the coordinates of B Prime and finally since point D is below the line of symmetry we'll start by counting how many units vertically it is away from the line y equals negative x and then repeat that distance moving horizontally above the line of symmetry now we have the coordinates of D Prime at five negative two by connecting those vertices we can struck the image of b-prime c-prime d-prime to create a very interesting picture we can rotate the graph to better visualize what this reflection looks like now we can compare the point B with coordinates 7 1 and B Prime at negative 1 negative 7 C remains unchanged and D with coordinates 2 negative 5 2 D Prime 5 negative 2 if you notice for this relationship when reflecting any point over the line y equals negative x the x and y values are switched and negated so XY becomes negative y negative x and now our final words on reflections reflection is a mirror image across a specified line of symmetry for these transformations you may apply reflection rules for common lines of symmetry these common lines of symmetry include the X and y axis the lines x equals K and y equals K and the line y equals x and y equals negative x thanks a lot guys you're checking out that lesson please subscribe to our YouTube channel and head on over to mash up matcom for more content and practice activities you can also file a mashup map on Instagram and Twitter for daily updates answer keys and more exclusive content so please hit us up and let us know what you think

Channel: Mashup Math

Views: 779,499

Keywords: geometry reflections y=x, geometry reflections video, khan academy geometry reflections, reflection across the x axis, reflection across the y axis, reflection across, y=x, y=-x, reflection over x axis, reflection over y axis, reflection over, reflect a point over a line, reflection in the x axis, reflection in the y axis, reflection in, khan academy, math antics, common core, reflection over y=2, reflection over x and y axis, reflect a figure in a horizontal or vertical line

Id: ouNp8FtgiEE

Length: 11min 25sec (685 seconds)

Published: Tue Mar 31 2015