# Lesson 4

Reflecting Functions

• Let’s reflect some graphs.

### 4.1: Notice and Wonder: Reflections

What do you notice? What do you wonder?

### 4.2: Reflecting Across

Here is the graph of function $$f$$ and a table of values.

$$x$$ $$f(x)$$ $$g(x) = \text-f(x)$$
-3 0
-1.5 -4.3
-1 -4
0 -1.8
0.6 0
2.6 3.9
4 0
1. Let $$g$$ be the function defined by $$g(x) = \text-f(x)$$. Complete the table.
2. Sketch the graph of $$g$$ on the same axes as the graph of $$f$$ but in a different color.
3. Describe how to transform the graph of $$f$$ into the graph of $$g$$. Explain how the equation produces this transformation.

### 4.3: Reflecting Across a Different Way

Here is another copy of the graph of $$f$$ from the earlier activity. This time, let $$h$$ be the function defined by $$h(x) = f(\text-x)$$.

1. Use the definition of $$h$$ to find $$h(0)$$. Does your answer agree with your prediction?
2. What does your prediction tell you about $$h(\text-0.6)$$? Does your answer agree with the definition of $$h$$?
3. Complete the tables. The values for $$x$$ will not be the same for the two tables.

$$x$$ $$f(x)$$
-3 0
-1.5 -4.3
-1 -4
0 -1.8
0.6 0
2.6 3.9
4 0
$$x$$ $$h(x)=f(\text-x)$$

4. Sketch the graph of $$h$$ on the same axes as the graph of $$f$$ but in a different color.
5. Describe what happened to the graph of $$f$$ to transform it into the graph of $$h$$. Explain how the equation produces this transformation.

1. Describe how the graph of $$h$$ relates to the graph of $$g$$ defined in the earlier activity.
2. Write an equation relating $$h$$ and $$g$$.

### Summary

Here are graphs of the functions $$f$$, $$g$$, and $$h$$, where $$g(x)=\text-f(x)$$ and $$h(x)=f(\text-x)$$. How do these equations match the transformation we see from $$f$$ to $$g$$ and from $$f$$ to $$h$$?

Considering first the equation $$g(x)=\text-f(x)$$, we know that for the same input $$x$$, the value of $$g(x)$$ will be the opposite of the value of $$f(x)$$. For example, since $$f(0)=1$$, we know that $$g(0)=\text-f(0)=\text-1$$. We can see this relationship in the graphs where $$g$$ is the reflection of $$f$$ across the $$x$$-axis.

Looking at $$h(x)=f(\text-x)$$, this equation tells us that the two functions have the same output for opposite inputs. For example, 1 and -1 are opposites, so $$h(1)=f(\text-1)$$ (and $$h(\text-1)=f(1)$$ is also true!). We can see this relationship in the graphs where $$h$$ is the reflection of $$f$$ across the $$y$$-axis.