Section 9.2 Applying Transformations
In Section 9.1, we learned how to determine what transformations were applied to a function. In this section, we will go backwards: we tell you what tranformations need to be applied, and you have to apply them to the function. It's important to note that Table 9.4 will still be very helpful in organizing the information! Specifically, you need to still think about three key things:Where does the transformation go/what direction is it going in? Vertical transformations are applied to the outputs, and horizontal transformations are applied to the inputs.
What kind of transformation/operation is being applied? Shifts involve adding or subtracting, scales involve multiplying or dividing by a positive number, and reflections involve multiplying by
Horizontal transformations act backwards from what we expect.
Example 9.9.
Suppose the graph of
When the question asks for the equation "in terms of"
So, the key to this problem is to figure out how each transformation changes the function notation:
-
Shifting up by 2:
Up is vertical, so this is applied to the outside.
Shifts are adding/subtracting.
We will add 2 to the outside of the function.
-
Stretching horizontally by 7:
Horizontally is applied to the inside.
Stretch is a scale, so we are multiplying or dividing.
Remember: horizontal transformations are backwards! So, we will divide the inputs by 7.
We put this all together to get the final answer:
Checkpoint 9.10.
Suppose the graph of
The key to this problem is to figure out how each transformation changes the function notation:
-
Shifting left by 5:
Left is horizontal, so this is applied to the inside.
Shifts are adding/subtracting.
Remember: horizontal transformations are backwards! So, we will add 5 to the inputs, since left is in the negative
-direction.
-
Vertical flip:
Vertical is applied to the outside.
Flipping involves multiplying by
So, we will multiply the outside by
We put this all together to get the final answer:
Example 9.11.
Suppose
Just like before, we need to figure out how each transformation changes the formula:
-
Shifting left by 2:
Left is horizontal, so this is applied to the inside.
Shifts are adding/subtracting.
Remember: horizontal transformations are backwards! So, we will add 2 to the inputs, since left is in the negative
-direction.
-
Shifting up by 9:
Vertical is applied to the outside.
Shifts are adding/subtracting.
So, we will add
to the outside.
At this point, we should write the function notation like we did when we didn't have an explicit formula. This makes it a lot easier to know how to change the formula later. In this case, our function notation is
Now that we know how the transformations affect the formula, we are ready to apply it to the formula we were given for
At this point, we are done with the question. Since we were told not to simplify our answer, we are not going to distribute any of the parentheses. We could combine the
Checkpoint 9.12.
Suppose
We need to figure out how each transformation changes the formula:
-
Vertical stretch by 2:
Vertical is applied to the outside.
Stretch is a scale, so we are multiplying/dividing.
We will multiply by 2 on the outside.
-
Horizontal stretch by 6:
Horizontal is applied to the inside.
Stretch is a scale, so we are multiplying/dividing.
Remember: horizontal transformations are backwards! So, to make the graph bigger horizontally, we will divide by 6 on the inside.
-
Vertical flip:
Vertical is applied to the outside.
Flipping involves multiplying by
So, we will multiply the outside by
At this point, we should write the function notation like we did when we didn't have an explicit formula. This makes it a lot easier to know how to change the formula later. In this case, our function notation is
Now that we know how the transformations affect the formula, we are ready to apply it to the formula we were given for