6.01 Reflections

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.

Reflection across the $x$x-axis

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. 

Reflection across the $y$y-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:

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.

Reflection across both axes

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. 

Summary

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.

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5