Lesson 4

What do you notice? What do you wonder?

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.

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.

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 $h$?

$f(x)$

$g(x)=\text-f(x)$

$h(x) = f(\text-x)$

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.