## Section6.1Reflections

To begin this activity, first an exercise about function notation.

We refer to the number $-5$ by the name “negative $5$”. The word negative always refers to things less than zero. So, what should we call the expression $-N\text{?}$

If $N$ is a variable (or an unknown quantity), then it may not make sense to say “negative $N$”. If $N$ is a positive number, then this would be fine, but if $N$ is a negative number, then $-N$ would be positive. No matter whether $N$ is positive or negative to begin with, the expression $-N$ will have the opposite sign of $N\text{.}$

Therefore, we will call $-N$ the opposite of $N\text{.}$

This is particularly useful when we don't know if a variable is positive or negative.

We refer to $-x$ as the opposite of $x\text{,}$ and we avoid saying negative $x\text{.}$

Check your understanding in the next exercise, where you will locate the opposite of a number.

In this activity, it will be important to know how to write an expression which represents the opposite of something else.

If $p(x) = x - 3$ then the opposite of $p(x)$ is:

\begin{align*} -p(x) \amp= -(x - 3)\\ \amp= -x + 3 \end{align*}

This represents the opposite of the output of $p(x)\text{.}$

In the next exercise, you will practice writing expressions involving the opposite.

### Subsection6.1.1The transformation $y = -f(x)$

Suppose $f$ is a function. If $f(-5)$ is a positive number, then $-f(-5)$ will be negative, and if $f(2)$ is a negative number, then $-f(2)$ will be positive.

The same input is used, but the outputs are opposites.

In general, whatever output values $f(x)$ has, the transformation

\begin{equation*} y = -f(x) \end{equation*}

will have the opposite output values.

See the following exercise for the graph of the transformation $y = -f(x)\text{.}$

Now use what you just saw in order to create the graph of $y = -f(x)\text{.}$

### Subsection6.1.2The transformation $y = f(-x)$

For a function $g(x)\text{,}$ the transformation $g(-x)$ does something else.

If you wanted to evaluate $g(-x)$ when $x = 4\text{,}$ you would really be evaluating $g(-4)\text{.}$ Or, if you wanted to evaluate $g(-x)$ when $x = -9\text{,}$ you would really be evaluating $g(9)\text{.}$

The transformation $g(-x)$ evaluates the function $g(x)$ at the opposite input.

See the next exercise to explore the transformation $y = g(-x)$ graphically.

Now use what you just saw in order to create the graph of $y = g(-x)\text{.}$

In the next exercise, you will evaluate $f(-x)$ and $-f(x)$ in a table of values. Remember that the opposite just changes the sign on a number, whether it is an input or an output.

### Subsection6.1.3Exercises

In the next two exercises, use function notation to describe how a graph was reflected, and then write the formula for a reflected function.

Finally, we discuss a method for how to graph the transformations $y = -f(x)$ and $y = f(-x)\text{.}$