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# 2.5: Using Transformations to Graph Functions

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Skills to Develop

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

## Vertical and Horizontal Translations

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.

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 $$\PageIndex{1}$$:

Sketch the graph of $$g(x)=\sqrt{x}+4$$.

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 $$\PageIndex{2}$$:

Sketch the graph of $$g(x)=(x−4)^{3}$$.

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 $$\PageIndex{3}$$:

Sketch the graph of $$g(x)=|x+3|−5$$.

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 $$\PageIndex{4}$$:

Sketch the graph of $$g ( x ) = \frac { 1 } { x - 5 } + 3$$.

Solution

Begin with the reciprocal function and identify the translations.

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

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 $$\PageIndex{1}$$

Sketch the graph of $$g ( x ) = ( x - 2 ) ^ { 2 } + 1$$.

## 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 $$\PageIndex{5}$$:

Sketch the graph of $$g ( x ) = - ( x + 5 ) ^ { 2 } + 3$$.

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 $$\PageIndex{2}$$

Sketch the graph of $$g ( x ) = - | x | + 3$$.

## 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 $$\PageIndex{6}$$:

Sketch the graph of $$g ( x ) = - 2 | x - 5 | - 3$$.

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.

Exercise $$\PageIndex{3}$$

Match the graph to the function definition.

1. $$f(x) = \sqrt{x + 4}$$
2. $$f(x) = |x − 2| − 2$$
3. $$f(x) = \sqrt{x + 1} -1$$
4. $$f(x) = |x − 2| + 1$$
5. $$f(x) = \sqrt{x + 4} + 1$$
6. $$f(x) = |x + 2| − 2$$

1. e

3. d

5. f

Exercise $$\PageIndex{4}$$

Graph the given function. Identify the basic function and translations used to sketch the graph. Then state the domain and range.

1. $$f(x) = x + 3$$
2. $$f(x) = x − 2$$
3. $$g(x) = x^{2} + 1$$
4. $$g(x) = x^{2} − 4$$
5. $$g(x) = (x − 5)^{2}$$
6. $$g(x) = (x + 1)^{2}$$
7. $$g(x) = (x − 5)^{2} + 2$$
8. $$g(x) = (x + 2)^{2} − 5$$
9. $$h(x) = |x + 4|$$
10. $$h(x) = |x − 4|$$
11. $$h(x) = |x − 1| − 3$$
12. $$h(x) = |x + 2| − 5$$
13. $$g(x) = \sqrt{x} − 5$$
14. $$g(x) = \sqrt{x − 5}$$
15. $$g(x) = \sqrt{x − 2} + 1$$
16. $$g(x) = \sqrt{x + 2} + 3$$
17. $$h(x) = (x − 2)^{3}$$
18. $$h(x) = x^{3} + 4$$
19. $$h(x) = (x − 1)^{3} − 4$$
20. $$h(x) = (x + 1)^{3} + 3$$
21. $$f(x) = \frac{1}{x−2}$$
22. $$f(x) = \frac{1}{x+3}$$
23. $$f(x) = \frac{1}{x} + 5$$
24. $$f(x) = \frac{1}{x} − 3$$
25. $$f(x) = \frac{1}{x+1} − 2$$
26. $$f(x) = \frac{1}{x−3} + 3$$
27. $$g(x) = −4$$
28. $$g(x) = 2$$
29. $$f ( x ) = \sqrt [ 3 ] { x - 2 } + 6$$
30. $$f ( x ) = \sqrt [ 3 ] { x + 8 } - 4$$

1. $$y = x$$; Shift up $$3$$ units; domain: $$\mathbb{R}$$; range: $$\mathbb{R}$$

3. $$y = x^{2}$$; Shift up $$1$$ unit; domain: $$ℝ$$; range: $$[1, ∞)$$

5. $$y = x^{2}$$; Shift right $$5$$ units; domain: $$ℝ$$; range: $$[0, ∞)$$

7. $$y = x^{2}$$; Shift right $$5$$ units and up $$2$$ units; domain: $$ℝ$$; range: $$[2, ∞)$$

9. $$y = |x|$$; Shift left $$4$$ units; domain: $$ℝ$$; range: $$[0, ∞)$$

11. $$y = |x|$$; Shift right $$1$$ unit and down $$3$$ units; domain: $$ℝ$$; range: $$[−3, ∞)$$

13. $$y = \sqrt{x}$$; Shift down $$5$$ units; domain: $$[0, ∞)$$; range: $$[−5, ∞)$$

15. $$y = \sqrt{x}$$; Shift right $$2$$ units and up $$1$$ unit; domain: $$[2, ∞)$$; range: $$[1, ∞)$$

17. $$y = x^{3}$$ ; Shift right $$2$$ units; domain: $$ℝ$$; range: $$ℝ$$

19. $$y = x^{3}$$; Shift right $$1$$ unit and down $$4$$ units; domain: $$ℝ$$; range: $$ℝ$$

21. $$y = \frac{1}{x}$$; Shift right $$2$$ units; domain: $$(−∞, 2) ∪ (2, ∞)$$; range: $$(−∞, 0) ∪ (0, ∞)$$

23. $$y = \frac{1}{x}$$; Shift up $$5$$ units; domain: $$(−∞, 0) ∪ (0, ∞)$$; range: $$(−∞, 1) ∪ (1, ∞)$$

25. $$y = \frac{1}{x}$$; Shift left $$1$$ unit and down $$2$$ units; domain: $$(−∞, −1) ∪ (−1, ∞)$$; range: $$(−∞, −2) ∪ (−2, ∞)$$

27. Basic graph $$y = −4$$; domain: $$ℝ$$; range: $$\{−4\}$$

29. $$y = \sqrt [ 3 ] { x }$$; Shift up $$6$$ units and right $$2$$ units; domain: $$ℝ$$; range: $$ℝ$$

Exercise $$\PageIndex{5}$$

Graph the piecewise functions.

1. $$h ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } + 2 } & { \text { if } x < 0 } \\ { x + 2 } & { \text { if } x \geq 0 } \end{array} \right.$$
2. $$h ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } - 3 \text { if } x < 0 } \\ { \sqrt { x } - 3 \text { if } x \geq 0 } \end{array} \right.$$
3. $$h ( x ) = \left\{ \begin{array} { l l } { x ^ { 3 } - 1 } & { \text { if } x < 0 } \\ { | x - 3 | - 4 } & { \text { if } x \geq 0 } \end{array} \right.$$
4. $$h ( x ) = \left\{ \begin{array} { c c } { x ^ { 3 } } & { \text { if } x < 0 } \\ { ( x - 1 ) ^ { 2 } - 1 } & { \text { if } x \geq 0 } \end{array} \right.$$
5. $$h ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } - 1 } & { \text { if } x < 0 } \\ { 2 } & { \text { if } x \geq 0 } \end{array} \right.$$
6. $$h ( x ) = \left\{ \begin{array} { l l } { x + 2 } & { \text { if } x < 0 } \\ { ( x - 2 ) ^ { 2 } } & { \text { if } x \geq 0 } \end{array} \right.$$
7. $$h ( x ) = \left\{ \begin{array} { l l } { ( x + 10 ) ^ { 2 } - 4 } & { \text { if } x < - 8 } \\ { x + 4 } & { \text { if } - 8 \leq x < - 4 } \\ { \sqrt { x + 4 } } & { \text { if } x \geq - 4 } \end{array} \right.$$
8. $$f ( x ) = \left\{ \begin{array} { l l } { x + 10 } & { \text { if } x \leq - 10 } \\ { | x - 5 | - 15 } & { \text { if } - 10 < x \leq 20 } \\ { 10 } & { \text { if } x > 20 } \end{array} \right.$$

1.

3.

5.

7.

Exercise $$\PageIndex{6}$$

Write an equation that represents the function whose graph is given.

1.

2.

3.

4.

5.

6.

7.

8.

1. $$f ( x ) = \sqrt { x - 5 }$$

3. $$f ( x ) = ( x - 15 ) ^ { 2 } - 10$$

5. $$f ( x ) = \frac { 1 } { x + 8 } + 4$$

7. $$f ( x ) = \sqrt { x + 16 } - 4$$

Exercise $$\PageIndex{6}$$

Match the graph to the given function defintion.

1. $$f ( x ) = - 3 | x |$$
2. $$f ( x ) = - ( x + 3 ) ^ { 2 } - 1$$
3. $$f ( x ) = - | x + 1 | + 2$$
4. $$f ( x ) = - x ^ { 2 } + 1$$
5. $$f ( x ) = - \frac { 1 } { 3 } | x |$$
6. $$f ( x ) = - ( x - 2 ) ^ { 2 } + 2$$

1. b

3. d

5. f

Exercise $$\PageIndex{7}$$

Use the transformations to graph the following functions.

1. $$f ( x ) = - x + 5$$
2. $$f ( x ) = - | x | - 3$$
3. $$g ( x ) = - | x - 1 |$$
4. $$f ( x ) = - ( x + 2 ) ^ { 2 }$$
5. $$h ( x ) = \sqrt { - x } + 2$$
6. $$g ( x ) = - \sqrt { x } + 2$$
7. $$g ( x ) = - ( x + 2 ) ^ { 3 }$$
8. $$h ( x ) = - \sqrt { x - 2 } + 1$$
9. $$g ( x ) = - x ^ { 3 } + 4$$
10. $$f ( x ) = - x ^ { 2 } + 6$$
11. $$f ( x ) = - 3 | x |$$
12. $$g ( x ) = - 2 x ^ { 2 }$$
13. $$h ( x ) = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 }$$
14. $$h ( x ) = \frac { 1 } { 3 } ( x + 2 ) ^ { 2 }$$
15. $$g ( x ) = - \frac { 1 } { 2 } \sqrt { x - 3 }$$
16. $$f ( x ) = - 5 \sqrt { x + 2 }$$
17. $$f ( x ) = 4 \sqrt { x - 1 } + 2$$
18. $$h ( x ) = - 2 x + 1$$
19. $$g ( x ) = - \frac { 1 } { 4 } ( x + 3 ) ^ { 3 } - 1$$
20. $$f ( x ) = - 5 ( x - 3 ) ^ { 2 } + 3$$
21. $$h ( x ) = - 3 | x + 4 | - 2$$
22. $$f ( x ) = - \frac { 1 } { x }$$
23. $$f ( x ) = - \frac { 1 } { x + 2 }$$
24. $$f ( x ) = - \frac { 1 } { x + 1 } + 2$$

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

Exercise $$\PageIndex{8}$$

1. Use different colors to graph the family of graphs defined by $$y=kx^{2}$$, where $$k \in \left\{ 1 , \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 } \right\}$$. What happens to the graph when the denominator of $$k$$ is very large? Share your findings on the discussion board.
2. Graph $$f ( x ) = \sqrt { x }$$ and $$g ( x ) = - \sqrt { x }$$ on the same set of coordinate axes. What does the general shape look like? Try to find a single equation that describes the shape. Share your findings.
3. Explore what happens to the graph of a function when the domain values are multiplied by a factor $$a$$ before the function is applied, $$f(ax)$$. Develop some rules for this situation and share them on the discussion board.

## 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.