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6.01 Reflections

Lesson

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(x2). 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:

$f\left(-x\right)$f(x) $=$= $-x\left(-x-2\right)$x(x2)

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.

 

 

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

Reflections of the function $y=f\left(x\right)$y=f(x)
$y=-f\left(x\right)$y=f(x) is a vertical reflection across the $x$x-axis: the sign of every $y$y-value on the original graph changes
$y=f\left(-x\right)$y=f(x) is a horizontal reflection across the $y$y-axis: the sign of every $x$x-value on the original graph changes
$y=-f\left(-x\right)$y=f(x) is a reflection across both the $x$x- and $y$y-axes, or, 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.

 

Practice questions

Question 1

Consider the following.

  1. 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)(x2)(x5).

    Give your answer in the form '$x=\text{. . .}$x=. . .'.

  2. 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)(x2)(x5) across the $x$x-axis.

    State the equation of this function.

Question 2

A graph of the function $f\left(x\right)=\frac{1}{x+6}-3$f(x)=1x+63 is shown below.

Loading Graph...

  1. 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 $\editable{}=\editable{}$=.

  2. Sketch a graph of the new function $f\left(-x\right)$f(x).

    Loading Graph...

Question 3

Consider the function $f\left(x\right)=x^2+3$f(x)=x2+3.

  1. Find an expression for the function $-f\left(x\right)$f(x), in simplest form.

  2. 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:

    Loading Graph...

  3. 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.

    A

    Rotation by $90^\circ$90° clockwise about the origin.

    B

    Reflection across the $x$x-axis.

    C

    Reflection across the $y$y-axis.

    D

Question 4

Consider the function $f\left(x\right)=\left|2x-4\right|$f(x)=|2x4|.

  1. Find an expression for the function $f\left(-x\right)$f(x), in simplest form.

  2. 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:

    Loading Graph...

  3. 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.

    A

    Reflection across the $y$y-axis.

    B

    Rotation by $90^\circ$90° anticlockwise about the origin.

    C

    Rotation by $90^\circ$90° clockwise about the origin.

    D

Question 5

From the graph of $f\left(x\right)=x^3-16x$f(x)=x316x, sketch the graph of $-f\left(-x\right)$f(x).

  1. Loading Graph...

Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

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