Graphing Functions Using Reflections

The next transformation we will learn is reflections. You will reflect graphs of functions across the axes and determine how the transformations change the equations of the functions.

Graphing Functions Using Reflections about the Axes

Another transformation that can be applied to a function is a reflection over the x - or y-axis. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y axis. The reflections are shown in Figure 12.

Figure 12 Vertical and horizontal reflections of a function.

Figure 12 Vertical and horizontal reflections of a function.


Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y-axis.


REFLECTIONS

Given a function f(x), a new function g(x)=-f(x) is a vertical reflection of the function f(x), sometimes called a reflection about (or over, or through) the x-axis.

Given a function f(x), a new function g(x)=f(-x) is a horizontal reflection of the function f(x), sometimes called a reflection about the y-axis.


HOW TO

Given a function, reflect the graph both vertically and horizontally.

  1. Multiply all outputs by -1 for a vertical reflection. The new graph is a reflection of the original graph about the x-axis.
  2. Multiply all inputs by -1 for a horizontal reflection. The new graph is a reflection of the original graph about the y-axis.


EXAMPLE 9

Reflecting a Graph Horizontally and Vertically

Reflect the graph of s(t)=\sqrt{t}(\mathrm{a}) vertically and (b) horizontally.


Solution

(a)

Reflecting the graph vertically means that each output value will be reflected over the horizontal t- axis as shown in Figure 13.


Figure 13 Vertical reflection of the square root function


Because each output value is the opposite of the original output value, we can write

V(t)=-s(t) \quad \text { or } V(t)=-\sqrt{t}

Notice that this is an outside change, or vertical shift, that affects the output s(t) values, so the negative sign belongs outside of the function.

b)

Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 14.


Figure 14 Horizontal reflection of the square root function


Because each input value is the opposite of the original input value, we can write

H(t)=s(-t) \quad \text { or } H(t)=\sqrt{-t}

Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.

Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [0, \infty) and range [0, \infty), the vertical reflection gives the V(t) function the range (-\infty, 0] and the horizontal reflection gives the H(t) function the domain (-\infty, 0].


TRY IT #5

Reflect the graph of f(x)=|x-1| (a) vertically and (b) horizontally.


EXAMPLE 10

Reflecting a Tabular Function Horizontally and Vertically

A function f(x) is given as Table 6. Create a table for the functions below.

(a) g(x)=-f(x)

(b) h(x)=f(-x)

x 2 4 6 8
f(x) 1 3 7 11
Table 6


Solution

(a)

For g(x), the negative sign outside the function indicates a vertical reflection, so the x-values stay the same and each output value will be the opposite of the original output value. See Table 7.

x 2 4 6 8
g(x) –1 –3 –7 –11
Table 7


(b)

For h(x), the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the h(x) values stay the same as the f(x) values. See Table 8.

x −2 −4 −6 −8
h(x) 1 3 7 11
Table 8


TRY IT #6

A function f(x) is given as Table 9. Create a table for the functions below.

(a) g(x)=-f(x)

(b) h(x)=f(-x)

x −2 0 2 4
f(x) 5 10 15 20
Table 9


EXAMPLE 11

Applying a Learning Model Equation

A common model for learning has an equation similar to k(t)=-2^{-t}+1, where k is the percentage of mastery that can be achieved after t practice sessions. This is a transformation of the function f(t)=2^{t} shown in Figure 15. Sketch a graph of k(t).


Figure 15


Solution

This equation combines three transformations into one equation.

  • A horizontal reflection: f(-t)=2^{-t}
  • A vertical reflection: -f(-t)=-2^{-t}
  • A vertical shift: -f(-t)+1=-2^{-t}+1

We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points (0, 1) and (1, 2).

  1. First, we apply a horizontal reflection: (0, 1) (–1, 2).
  2. Then, we apply a vertical reflection: (0, -1) (-1, –2)
  3. Finally, we apply a vertical shift: (0, 0) (-1, -1)).

This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.

In Figure 16, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.


Figure 16


Analysis

As a model for learning, this function would be limited to a domain of t \geq 0, with corresponding range [0,1).


TRY IT #7

Given the toolkit function f(x)=x^{2}, graph g(x)=-f(x) and h(x)=f(-x). Take note of any surprising behavior for these functions.



Source: Rice University, https://openstax.org/books/college-algebra/pages/3-5-transformation-of-functions
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