# Precalculus : Reflections

## Example Questions

### Example Question #1 : Transformations

Reflect  across , and then across . What is the equation of the line after the reflections?

Explanation:

The distance between the lines  and  is 6 units. Therefore, the first reflection should be 6 units below .  Therefore, the first reflection is:

The distance between  and  is 7 units.  Therefore, second reflection should be 7 units above .

The result is .

### Example Question #2 : Transformations

Reflect  across .  Reflect again across .  What is the equation of the line after both reflections?

Explanation:

The result of reflecting  across  will become .  Reflecting the equation  across  will become .

### Example Question #2 : Transformations

If  was reflected across , then reflected across , and then reflected again across , what is the equation after the three reflections?

Explanation:

First, if  was reflected across , the new line will be .

Reflecting this line  across  will yield .

Reflecting  across  will yield .

### Example Question #1 : Reflections

Which of these functions has been shifted one unit to the right, three units down, and reflected across the -axis?

Explanation:

So far all of the transformations have come after or inside the function. But what about those that come before the function? This is the basis of reflections which can be made to reflect across the x or the y axis. Since a function takes an x value and returns a y value, the placement of our sign matters. For instance,  is a parabola opening up whose vertex is at the origin. But if one was to place a  -  sign in front of the fucntion like this , then the function would always return a negative y value. Because of this, a negative sign in front of the function, but not "in" it will reflect over the x axis. This is equivalent to turning the first function,,  a smile, into a frown. Use a graphing utility to visualize this. Now what if you placed a negative sign within the function? This would automatically make all x values negative. And since a function returns a y value and we will always be starting from the negative side of the x axis, this would reflect across the y axis. Use the function  to visualize this in a graphing utility.

is the correct answer because a negative 1 inside the function (which is within the square root) shifts the function to the right (it is opposite of intuition) one unit and outside the function a negative 3 shifts the function 3 units down. Finally, because our negative sign is outside the function we will always yield negative y values and have thus reflected over the x axis.