Transformations on Functions

A transformation simply refers to moving the graph of a function around. If you want to move the graph of y = x + 1:

• left or right - increase or decrease the input, which is usually x.
• up or down, you increase or decrease the output, which is usually y.

Normally, the input of 2 for y = x + 1 will have the output of 3 and my xy‐coordinate is (2, 3). However, if I decrease the input by 1, y = (x – 1) + 1, then my output will be 2 and my xy‐coordinate is (2, 2). The resulting line graph will move to the right by one interval.

See the following table of transformations, which shows moving the function up, down, left, and right by 1 unit, as well a reflection of the function over the x-axis:

Examine the following equation and see how it is shifted horizontally to the right by one interval.

• y = 2x + 1
• y = 2(x - 1) + 1

You can determine how the transformed equation will look by working the equation out.

• y = 2(x - 1) + 1 = 2x - 2 + 1 = 2x - 1

We can now evaluate the differences between the original equation, y = 2x + 1, and the shifted equation, y = 2x - 1.

You may notice that a horizontal shift can also be seen as a vertical shift for lines! This is something to keep in mind as you transform a linear equation to the left or right.

Practice Problems

Answers to Practice Problems

1. 0, 2, 4, 6 are the outputs for 1, 2, 3, and 4, respectively.
2. -1, 1, 3, 5 are the outputs for 1, 2, 3, and 4, respectively.
3. 12, 14, 16, 18 are the outputs for 1, 2, 3, and 4, respectively.
4. 4, 6, 8, 10 are the outputs for 1, 2, 3, and 4, respectively.
5. B
6. A
7. C
8. C