# 4.3: Understanding Transformations of Functions

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Learning Objectives

• Define the rigid transformations and use them to sketch graphs.
• Define the non-rigid transformations and use them to sketch graphs.

Prerequisite Skills

Before you get started, take this prerequisite quiz.

1. Sketch a graph of each of the following functions.  Include at least 3 key points on each graph.

a. $$y=x$$

b. $$y=x^2$$

c. $$y=x^3$$

d. $$y=|x|$$

e. $$y=\sqrt{x}$$

f. $$y=\dfrac{1}{x}$$

If you missed any part of this problem, review Section 4.2. (Note that this will open in a new window.)

When the graph of a function is changed in appearance and/or location we call it a transformation. There are two types of transformations. A rigid transformation57 changes the location of the function in a coordinate plane, but leaves the size and shape of the graph unchanged. A non-rigid transformation58 changes the size and/or shape of the graph.

## Vertical and Horizontal Translations

A vertical translation59 is a rigid transformation that shifts a graph up or down relative to the original graph. This occurs when a constant is added to any function. If we add a positive constant to each $$y$$-coordinate, the graph will shift up. If we add a negative constant, the graph will shift down. For example, consider the functions $$g(x) = x^{2} − 3$$ and $$h(x) = x^{2} + 3$$. Begin by evaluating for some values of the independent variable $$x$$.

Now plot the points and compare the graphs of the functions $$g$$ and $$h$$ to the basic graph of $$f(x) = x^{2}$$, which is shown using a dashed grey curve below.

The function $$g$$ shifts the basic graph down $$3$$ units and the function $$h$$ shifts the basic graph up $$3$$ units. In general, this describes the vertical translations; if $$k$$ is any positive real number:

Vertical shift up $$k$$ units: $$F(x)=f(x)+k$$ $$F(x)=f(x)-k$$

Table 2.5.1

Example $$\PageIndex{1}$$:

Identify the basic function and the transformation in $$g(x)=\sqrt{x}+4$$.  Then sketch the graph.

Solution

Begin with the basic function defined by $$f(x)=\sqrt{x}$$ and shift the graph up $$4$$ units.

A horizontal translation60 is a rigid transformation that shifts a graph left or right relative to the original graph. This occurs when we add or subtract constants from the $$x$$-coordinate before the function is applied. For example, consider the functions defined by $$g(x)=(x+3)^{2}$$ and $$h(x)=(x−3)^{2}$$ and create the following tables:

Here we add and subtract from the x-coordinates and then square the result. This produces a horizontal translation.

Note that this is the opposite of what you might expect. In general, this describes the horizontal translations; if $$h$$ is any positive real number:

Horizontal shift left $$h$$ units: $$F(x)=f(x+h)$$ $$F(x)=f(x-h)$$

Table 2.5.2

Example $$\PageIndex{2}$$:

Identify the basic function and the transformation in $$g(x)=(x−4)^{3}$$.  Then sketch the graph .

Solution

Begin with a basic cubing function defined by $$f(x)=x^{3}$$ and shift the graph $$4$$ units to the right.

It is often the case that combinations of translations occur.

Example $$\PageIndex{3}$$:

Identify the basic function and the transformation in $$g(x)=|x+3|−5$$.  Then sketch the graph.

Solution

$$\begin{array} { l } { y = | x | } \quad\quad\quad\quad\color{Cerulean}{Basic \:function} \\ { y = | x + 3 | } \quad\: \quad\color{Cerulean}{Horizontal \:shift \: left \:3 \:units} \\ { y = | x + 3 | - 5 } \:\:\:\color{Cerulean}{Vertical \:shift \:down \:5 \:units} \end{array}$$

The order in which we apply horizontal and vertical translations does not affect the final graph.

Example $$\PageIndex{4}$$:

Identify the basic function and the transformation in $$g ( x ) = \frac { 1 } { x - 5 } + 3$$.  Then sketch the graph.

Solution

Begin with the reciprocal function and identify the translations.

$$\begin{array} { l } { y = \frac{1}{x} } \quad\quad\quad\quad\color{Cerulean}{Basic \:function} \\ { y = \frac{1}{x-5} } \quad\: \quad\:\:\:\color{Cerulean}{Horizontal \:shift \: right \:5 \:units} \\ { y = \frac{1}{x-5} +3 } \:\:\:\:\:\:\:\color{Cerulean}{Vertical \:shift \:up \:3 \:units} \end{array}$$

Take care to shift the vertical asymptote from the y-axis 5 units to the right and shift the horizontal asymptote from the x-axis up 3 units.

Exercise $$\PageIndex{1}$$

Identify the basic function and the transformation in $$g ( x ) = ( x - 2 ) ^ { 2 } + 1$$.  Then sketch the graph.

## Reflections

A reflection61 is a transformation in which a mirror image of the graph is produced about an axis. In this section, we will consider reflections about the $$x$$- and $$y$$-axis. The graph of a function is reflected about the $$x$$-axis if each $$y$$-coordinate is multiplied by $$−1$$. The graph of a function is reflected about the $$y$$-axis if each $$x$$-coordinate is multiplied by $$−1$$ before the function is applied. For example, consider $$g(x)=\sqrt{−x}$$ and $$h(x)=−\sqrt{x}$$.

Compare the graph of $$g$$ and $$h$$ to the basic square root function defined by $$f(x)=\sqrt{x}$$, shown dashed in grey below:

The first function $$g$$ has a negative factor that appears “inside” the function; this produces a reflection about the $$y$$-axis. The second function $$h$$ has a negative factor that appears “outside” the function; this produces a reflection about the $$x$$-axis. In general, it is true that:

Reflection about the $$y$$-axis: $$F ( x ) = f ( - x )$$ $$F ( x ) = - f ( x )$$

Table 2.5.3

When sketching graphs that involve a reflection, consider the reflection first and then apply the vertical and/or horizontal translations.

Example $$\PageIndex{5}$$:

Identify the basic function and the transformation in $$g ( x ) = - ( x + 5 ) ^ { 2 } + 3$$.  Then sketch the graph.

Solution

Begin with the squaring function and then identify the transformations starting with any reflections.

$$\begin{array} { l } { y = x ^ { 2 } } \quad\quad\quad\quad\quad\quad\color{Cerulean}{Basic\: function.} \\ { y = - x ^ { 2 } } \quad\quad\quad\quad\quad\:\color{Cerulean}{Relfection\: about\: the\: x-axis.} \\ { y = - ( x + 5 ) ^ { 2 } } \quad\quad\:\:\:\color{Cerulean}{Horizontal\: shift\: left\: 5\: units.} \\ { y = - ( x + 5 ) ^ { 2 } + 3 } \quad\color{Cerulean}{Vertical\: shift\: up\: 3\: units.} \end{array}$$

Use these translations to sketch the graph.

Exercise $$\PageIndex{2}$$

Identify the basic function and the transformation in $$g ( x ) = - | x | + 3$$.  Then sketch the graph.

## Dilations

Horizontal and vertical translations, as well as reflections, are called rigid transformations because the shape of the basic graph is left unchanged, or rigid. Functions that are multiplied by a real number other than $$1$$, depending on the real number, appear to be stretched vertically or stretched horizontally. This type of non-rigid transformation is called a dilation62. For example, we can multiply the squaring function $$f(x) = x^{2}$$ by $$4$$ and $$\frac{1}{4}$$ to see what happens to the graph.

Compare the graph of $$g$$ and $$h$$ to the basic squaring function defined by $$f(x)=x^{2}$$, shown dashed in grey below:

The function $$g$$ is steeper than the basic squaring function and its graph appears to have been stretched vertically. The function $$h$$ is not as steep as the basic squaring function and appears to have been stretched horizontally.

In general, we have:

Dilation: $$F ( x ) = a \cdot f ( x )$$

Table 2.5.4

If the factor $$a$$ is a nonzero fraction between $$−1$$ and $$1$$, it will stretch the graph horizontally. Otherwise, the graph will be stretched vertically. If the factor $$a$$ is negative, then it will produce a reflection as well.

Example $$\PageIndex{6}$$:

Identify the basic function and the transformation in $$g ( x ) = - 2 | x - 5 | - 3$$.  Then sketch the graph.

Solution

Here we begin with the product of $$−2$$ and the basic absolute value function: $$y=−2|x|$$.This results in a reflection and a dilation.

Use the points $$\{(−1, −2), (0, 0), (1, −2)\}$$ to graph the reflected and dilated function $$y=−2|x|$$. Then translate this graph $$5$$ units to the right and $$3$$ units down.

$$\begin{array} { l } { y = - 2 | x | } \quad\quad\quad\quad\:\color{Cerulean}{Basic\: graph\: with\: dilation\: and\: reflection\: about\: the\: x-axis.}\\ { y = - 2 | x - 5 | } \quad\quad\:\:\color{Cerulean}{Shift\: right\: 5\: units.} \\ { y = - 2 | x - 5 | - 3 } \:\:\:\:\color{Cerulean}{Shift\: down\: 3\: units.} \end{array}$$

In summary, given positive real numbers $$h$$ and $$k$$:

Vertical shift up $$k$$ units: $$F(x)=f(x)+k$$ $$F(x)=f(x)-k$$

Table 2.5.1

Horizontal shift left $$h$$ units: $$F(x)=f(x+h)$$ $$F(x)=f(x-h)$$

Table 2.5.2

Reflection about the $$y$$-axis: $$F ( x ) = f ( - x )$$ $$F ( x ) = - f ( x )$$

Table 2.5.3

Dilation: $$F ( x ) = a \cdot f ( x )$$

Table 2.5.4

## Key Takeaways

• Identifying transformations allows us to quickly sketch the graph of functions. This skill will be useful as we progress in our study of mathematics. Often a geometric understanding of a problem will lead to a more elegant solution.
• If a positive constant is added to a function, $$f(x) + k$$, the graph will shift up. If a positive constant is subtracted from a function, $$f(x) − k$$, the graph will shift down. The basic shape of the graph will remain the same.
• If a positive constant is added to the value in the domain before the function is applied, $$f(x + h)$$, the graph will shift to the left. If a positive constant is subtracted from the value in the domain before the function is applied, $$f(x − h)$$, the graph will shift right. The basic shape will remain the same.
• Multiplying a function by a negative constant, $$−f(x)$$, reflects its graph in the $$x$$-axis. Multiplying the values in the domain by $$−1$$ before applying the function, $$f(−x)$$, reflects the graph about the $$y$$-axis.
• When applying multiple transformations, apply reflections first.
• Multiplying a function by a constant other than $$1$$, $$a ⋅ f(x)$$, produces a dilation. If the constant is a positive number greater than $$1$$, the graph will appear to stretch vertically. If the positive constant is a fraction less than $$1$$, the graph will appear to stretch horizontally.

Footnotes

57A set of operations that change the location of a graph in a coordinate plane but leave the size and shape unchanged.

58A set of operations that change the size and/or shape of a graph in a coordinate plane.

59A rigid transformation that shifts a graph up or down.

60A rigid transformation that shifts a graph left or right.

61A transformation that produces a mirror image of the graph about an axis.

62A non-rigid transformation, produced by multiplying functions by a nonzero real number, which appears to stretch the graph either vertically or horizontally.

4.3: Understanding Transformations of Functions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.