Section 6.1 Reflections
¶To begin this activity, first an exercise about function notation.
Checkpoint 6.1.1.
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.
Checkpoint 6.1.2.
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.
Checkpoint 6.1.3.
In this activity, it will be important to know how to write an expression which represents the opposite of something else.
Example 6.1.4.
If \(p(x) = x - 3\) then the opposite of \(p(x)\) is:
This represents the opposite of the output of \(p(x)\text{.}\)
In the next exercise, you will practice writing expressions involving the opposite.
Checkpoint 6.1.5.
Subsection 6.1.1 The 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
will have the opposite output values.
Checkpoint 6.1.6.
See the following exercise for the graph of the transformation \(y = -f(x)\text{.}\)
Checkpoint 6.1.7.
Now use what you just saw in order to create the graph of \(y = -f(x)\text{.}\)
Checkpoint 6.1.8.
Subsection 6.1.2 The 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.
Checkpoint 6.1.9.
See the next exercise to explore the transformation \(y = g(-x)\) graphically.
Checkpoint 6.1.10.
Now use what you just saw in order to create the graph of \(y = g(-x)\text{.}\)
Checkpoint 6.1.11.
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.
Checkpoint 6.1.12.
Subsection 6.1.3 Exercises
In the next two exercises, use function notation to describe how a graph was reflected, and then write the formula for a reflected function.
Checkpoint 6.1.13.
Checkpoint 6.1.14.
Finally, we discuss a method for how to graph the transformations \(y = -f(x)\) and \(y = f(-x)\text{.}\)