We have touched on the idea of reflection when considering the symmetry of odd and even functions. Remember that even functions, which are symmetrical across the $y$y-axis, have the property $f\left(x\right)=f\left(-x\right)$f(x)=f(−x), while odd functions have rotational symmetry around the origin since $f\left(x\right)=-f\left(-x\right)$f(x)=−f(−x).
We are in fact able to reflect a function across either the $x$x-axis or the $y$y-axis. We achieve this by multiplying the $y$y-values (the outputs of the function) or the $x$x-values (the inputs of the function) by $-1$−1 respectively.
Let's start with the example of $f\left(x\right)=x\left(x-2\right)$f(x)=x(x−2). This is a concave up parabola with two $x$x intercepts at $x=0$x=0 and $x=2$x=2. To reflect this across the $x$x-axis, we simply need to turn this into a concave down parabola. We do this by multiplying the all of the function values (commonly $y$y-values) by $-1$−1. In other words, the graph of $y=-f\left(x\right)$y=−f(x) is a reflection of $y=f\left(x\right)$y=f(x) across the $x$x-axis.
If a reflection across the $x$x-axis occurs when we multiply $y$y-values by $-1$−1, it is logical to assume that we will reflect across the $y$y-axis by multiplying the $x$x-values by $-1$−1. This is the same as finding $f\left(-x\right)$f(−x). Let's see how this works for our previous example:
$f\left(-x\right)$f(−x) | $=$= | $-x\left(-x-2\right)$−x(−x−2) |
factorising out the negative from the bracket |
$=$= | $x\left(x+2\right)$x(x+2) |
|
And we can see from the curve sketched below that $y=f\left(-x\right)$y=f(−x) is indeed a reflection of $y=f\left(x\right)$y=f(x) across the $y$y-axis.
Finally, we can combine both reflections if we calculate $y=-f\left(-x\right)$y=−f(−x). This will mean both the $x$x- and $y$y-values of every point on the original function will change from positive to negative, or vice versa. This is the same as creating a $180^\circ$180° rotation around the origin.
With this knowledge, we can answer both algebraic and graphical style questions. We need to be able to interpret function notation such as $y=f\left(-x\right)$y=f(−x) as an instruction to sketch a given graph's reflection across the $y$y-axis. We can use the location of the intercepts and turning points to create an accurate sketch. You might also be asked to match the correct function notation to a given reflection sketch.
Consider the following.
Determine the $x$x-intercepts of the cubic function $y=\left(x+3\right)\left(x-2\right)\left(x-5\right)$y=(x+3)(x−2)(x−5).
Give your answer in the form '$x=\text{. . .}$x=. . .'.
A second cubic function has the same $x$x-intercepts, but is a reflection of $y=\left(x+3\right)\left(x-2\right)\left(x-5\right)$y=(x+3)(x−2)(x−5) across the $x$x-axis.
State the equation of this function.
A graph of the function $f\left(x\right)=\frac{1}{x+6}-3$f(x)=1x+6−3 is shown below.
For the related function $f\left(-x\right)$f(−x), state the equations of the asymptotes:
The horizontal asymptote has equation $\editable{}=\editable{}$=.
The vertical asymptote has equation $\editable{}=\editable{}$=.
Sketch a graph of the new function $f\left(-x\right)$f(−x).
Consider the function $f\left(x\right)=x^2+3$f(x)=x2+3.
Find an expression for the function $-f\left(x\right)$−f(x), in simplest form.
A graph of $f\left(x\right)$f(x) is shown below. Sketch a graph of $-f\left(x\right)$−f(x) on the same axes:
Which of the following describes the transformation from the function $f\left(x\right)$f(x) to the function $-f\left(x\right)$−f(x)?
Rotation by $90^\circ$90° anticlockwise about the origin.
Rotation by $90^\circ$90° clockwise about the origin.
Reflection across the $x$x-axis.
Reflection across the $y$y-axis.
Consider the function $f\left(x\right)=\left|2x-4\right|$f(x)=|2x−4|.
Find an expression for the function $f\left(-x\right)$f(−x), in simplest form.
A graph of $f\left(x\right)$f(x) is shown below. Sketch a graph of $f\left(-x\right)$f(−x) on the same axes:
Which of the following describes the transformation from the function $f\left(x\right)$f(x) to the function $f\left(-x\right)$f(−x)?
Reflection across the $x$x-axis.
Reflection across the $y$y-axis.
Rotation by $90^\circ$90° anticlockwise about the origin.
Rotation by $90^\circ$90° clockwise about the origin.
From the graph of $f\left(x\right)=x^3-16x$f(x)=x3−16x, sketch the graph of $-f\left(-x\right)$−f(−x).