# Precalculus : Graphing Functions

## Example Questions

### Example Question #1 : Reflections

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 : Reflections

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 #3 : Reflections

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 #4 : 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.

### Example Question #1 : Determine The Symmetry Of An Equation

Is the following function symmetric across the y-axis? (Is it an even function?)

This isn't even a function!

Yes

No

Cannot be determined from the information given

No

Explanation:

One way to determine algebraically if a function is an even function, or symmetric about the y-axis, is to substitute  in for . When we do this, if the function is equivalent to the original, then the function is an even function. If not, it is not an even function.

For our function:

Thus the function is not symmetric about the y-axis.

### Example Question #2 : Determine The Symmetry Of An Equation

Is the following function symmetric across the y-axis? (Is it an even function?)

Yes

I don't know!

No

There is not enough information to determine

That's not a function!

Yes

Explanation:

One way to determine algebraically if a function is an even function, or symmetric about the y-axis, is to substitute  in for . When we do this, if the function is equivalent to the original, then the function is an even function. If not, it is not an even function.

For our function:

Since this matches the original, our function is symmetric across the y-axis.

### Example Question #3 : Determine The Symmetry Of An Equation

Determine if there is symmetry with the equation  to the -axis and the method used to determine the answer.

Explanation:

In order to determine if there is symmetry about the x-axis, replace all  variables with .   Solving for , if the new equation is the same as the original equation, then there is symmetry with the x-axis.

Since the original and new equations are not equivalent, there is no symmetry with the x-axis.

### Example Question #4 : Determine The Symmetry Of An Equation

Is the following function symmetrical about the y axis (is it an even function)?

Insufficient Information

Not a function

No

Yes

No

Explanation:

For a function to be even, it must satisfy the equality

Likewise if a function is even, it is symmetrical about the y-axis

Therefore, the function is not even, and so the answer is No

### Example Question #5 : Determine The Symmetry Of An Equation

Algebraically check for symmetry with respect to the x-axis, y axis, and the origin.

No symmetry

Explanation:

For a function to be symmetrical about the y-axis, it must satisfy   so there is not symmetry about the y-axis

For a function to be symmetrical about the x-axis, it must satisfy   so there is symmetry about the x-axis

For a function to be symmetrical about the origin, you must replace y with (-y) and x with (-x) and the resulting function must be equal to the original function.

### Example Question #6 : Determine The Symmetry Of An Equation

Algebraically check for symmetry with respect to the x-axis, y axis, and the origin.

Symmetry about the y-axis and origin

Symmetry about the x-axis, and y-axis

Symmetry about the x-axis, y-axis, and origin

Explanation:

For a function to be symmetrical about the y-axis, it must satisfy  so there is symmetry about the y-axis

For a function to be symmetrical about the x-axis, it must satisfy

so there is not symmetry about the x-axis

For a function to be symmetrical about the origin, you must replace y with (-y) and x with (-x) and the resulting function must be equal to the original function.

So there is no symmetry about the origin.