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Wize High School Grade 11 Math Textbook > Transformations of Functions

Reflections

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Vertical Reflections

Ify=f(x), y=f(x),~then y=−f(x) y=-f(x)~is a vertical reflection about the x-axis.

Example
Let us look at y=x2 & y=−x2 y=x^2~\&~y=-x^2~on the same grid:

If we compare the table of values for both functions
xx2−x2−24−4−11−100011−124−4\begin{array}{|c|c|c|} \hline x&x^2&-x^2\\\hline -2&4&-4\\ -1&1&-1\\ 0&0&0\\ 1&1&-1\\ 2&4&-4\\\hline \end{array}
we can see that all the output values of the parent function have been changed from positive to negative.
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Horizontal Reflections

If y=f(x)y=f(x), then y=f(−x)y=f(-x) is a horizontal reflection about the y-axis.


Example

Let us look at f(x)=(x−2)2 & g(x)=((−x)−2)2f(x)=(x-2)^2~\&~g(x)=((-x)-2)^2 on the same grid & their table of values:




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When we look at the table of values for f(x)=(x−2)2f(x)=(x-2)^2
x(x−2)2−216−19041120\begin{array}{|c|c|} \hline x&(x-2)^2\\\hline -2&16\\ -1&9\\ 0&4\\ 1&1\\ 2&0\\\hline \end{array}
And compare it to the table of values for g(x)=((−x)−2)2g(x)=((-x)-2)^2
x((−x)−2)22161904−11−20\begin{array}{|c|c|} \hline x&((-x)-2)^2\\\hline 2&16\\ 1&9\\ 0&4\\ -1&1\\ -2&0\\\hline \end{array}
we can see that the input values have changed signs indicating a horizontal reflection.
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Example: Vertical & Horizontal Reflections

Graph y=−x2y=-x^2 , then identifying the following:
  1. The parent function
  2. The table of values for parent function
  3. The table of values for transformed function
  4. Domain
  5. Range
The graph of the transformed function y=−x2y=-x^2 is:

Parent function: y=x2y=x^2

The table of values for the parent function:
xy−24−11001124\begin{array}{|c| c|} \hline x&y\\\hline -2&4\\ -1&1\\ 0&0\\ 1&1\\ 2&4\\\hline \end{array}

The table of values for the transformed function:
xy−2−4−1−1001−12−4\begin{array}{|c|c|}\hline x&y\\\hline -2&-4\\ -1&-1\\ 0&0\\ 1&-1\\ 2&-4\\\hline \end{array}
Domain: {x∈R∣ −∞<x<∞}\{x\in\mathbb{R}|~-\infin<x<\infin\}

Range: {y∈R∣ −∞<y<∞}\{y\in\mathbb{R}|~-\infin<y<\infin\}
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Example: Vertical & Horizontal Reflections

Sketch the transformed graph of the function, y=f(x), y=f(x),~(graphed below), stating, and identifying the axis of reflection:
  1. y=−f(x)y=-f(x)
  2. y=f(−x)y=f(-x)
  3. y=−f(−x)y=-f(-x)

Part A

y=−f(x)y=-f(x)

Sketch:

There is a vertical reflection about the x-axis

Part B

y=f(−x)y=f(-x)

Sketch:

There is a horizontal reflection about the y-axis

Part C

Sketch:

There is a vertical and horizontal reflection about the x-axis & y-axis respectively

Practice: Vertical & Horizontal Reflections

The point (3, -4) is on the graph of y=f(x).y=f(x).

Match the point to the appropriate transformed function.
A.
(3,4)(3, 4)
B.
(−3,−4)(-3, -4)
C.
(−3,4)(-3, 4)
y=−f(x)y=-f(x)
y=f(−x)y=f(-x)
y=−f(−x)y=-f(-x)

Practice: Vertical & Horizontal Reflections

Which of the following is the graph of f(x)=−−xf(x) = -\sqrt{-x}?

Practice: Vertical & Horizontal Reflections

The function y=(x−1)2+1 y=(x-1)^2+1~ has been vertically reflected about the x-axis and horizontally reflected about the y-axis.

Write the equation of the transformed function.