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Section6.1Reflections

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

The Opposite

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.

Example6.1.4

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.

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 \(y = f(-x)\text{.}\)