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2.5 Using Transformations to Graph Functions

Learning Objectives

  1. Define the rigid transformations and use them to sketch graphs.
  2. 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 transformationA set of operations that change the location of a graph in a coordinate plane but leave the size and shape unchanged. changes the location of the function in a coordinate plane, but leaves the size and shape of the graph unchanged. A non-rigid transformationA set of operations that change the size and/or shape of a graph in a coordinate plane. changes the size and/or shape of the graph.

A vertical translationA rigid transformation that shifts a graph up or down. 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)=x23 and h(x)=x2+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)=x2, 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

Vertical shift down k units:

F(x)=f(x)k

Example 1

Sketch the graph of g(x)=x+4.

Solution:

Begin with the basic function defined by f(x)=x and shift the graph up 4 units.

Answer:

A horizontal translationA rigid transformation that shifts a graph left or right. 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)=(x3)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)

Horizontal shift right h units:

F(x)=f(xh)

Example 2

Sketch the graph of g(x)=(x4)3.

Solution:

Begin with a basic cubing function defined by f(x)=x3 and shift the graph 4 units to the right.

Answer:

It is often the case that combinations of translations occur.

Example 3

Sketch the graph of g(x)=|x+3|5.

Solution:

Start with the absolute value function and apply the following transformations.

y=|x|Basicfunctiony=|x+3|Horizontalshiftleft3unitsy=|x+3|5Verticalshiftdown5units

Answer:

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

Example 4

Sketch the graph of g(x)=1x5+3.

Solution:

Begin with the reciprocal function and identify the translations.

y=1xBasicfunctiony=1x5Horizontalshiftright5unitsy=1x5+3Verticalshiftup3units

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.

Answer:

Try this! Sketch the graph of g(x)=(x2)2+1.

Answer:

Reflections

A reflectionA transformation that produces a mirror image of the graph about an axis. 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)=x and h(x)=x.

Compare the graph of g and h to the basic square root function defined by f(x)=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)

Reflection about the x-axis:

F(x)=f(x)

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

Example 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.

y=x2Basicfunction.y=x2Reflectionaboutthex-axis.y=(x+5)2Horizontalshiftleft5units.y=(x+5)2+3Verticalshiftup3units.

Use these translations to sketch the graph.

Answer:

Try this! Sketch the graph of g(x)=|x|+3.

Answer:

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 dilationA non-rigid transformation, produced by multiplying functions by a nonzero real number, which appears to stretch the graph either vertically or horizontally.. For example, we can multiply the squaring function f(x)=x2 by 4 and 14 to see what happens to the graph.

Compare the graph of g and h to the basic squaring function defined by f(x)=x2, 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)=af(x)

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 6

Sketch the graph of g(x)=2|x5|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.

xyy=2|x|Dilationandreflection12y=2|1|=21=200y=2|0|=20=012y=2|1|=21=2

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.

y=2|x|Basicgraphwithdilationandreflectionaboutthexaxis.y=2|x5|Shiftright5units.y=2|x5|3Shiftdown3units.

Answer:

In summary, given positive real numbers h and k:

Vertical shift up k units:

F(x)=f(x)+k

Vertical shift down k units:

F(x)=f(x)k

Horizontal shift left h units:

F(x)=f(x+h)

Horizontal shift right h units:

F(x)=f(xh)

Reflection about the y-axis:

F(x)=f(x)

Reflection about the x-axis:

F(x)=f(x)

Dilation:

F(x)=af(x)

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(xh), 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, af(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.

Topic Exercises

    Part A: Vertical and Horizontal Translations

    Match the graph to the function definition.

    1. f(x)=x+4

    2. f(x)=|x2|2

    3. f(x)=x+11

    4. f(x)=|x2|+1

    5. f(x)=x+4+1

    6. f(x)=|x+2|2

      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)=x2

    3. g(x)=x2+1

    4. g(x)=x24

    5. g(x)=(x5)2

    6. g(x)=(x+1)2

    7. g(x)=(x5)2+2

    8. g(x)=(x+2)25

    9. h(x)=|x+4|

    10. h(x)=|x4|

    11. h(x)=|x1|3

    12. h(x)=|x+2|5

    13. g(x)=x5

    14. g(x)=x5

    15. g(x)=x2+1

    16. g(x)=x+2+3

    17. h(x)=(x2)3

    18. h(x)=x3+4

    19. h(x)=(x1)34

    20. h(x)=(x+1)3+3

    21. f(x)=1x2

    22. f(x)=1x+3

    23. f(x)=1x+5

    24. f(x)=1x3

    25. f(x)=1x+12

    26. f(x)=1x3+3

    27. g(x)=4

    28. g(x)=2

    29. f(x)=x23+6

    30. f(x)=x+834

      Graph the piecewise functions.

    1. h(x)={x2+2ifx<0x+2ifx0
    2. h(x)={x23ifx<0x3ifx0
    3. h(x)={x31ifx<0|x3|4ifx0
    4. h(x)={ x3ifx<0(x1)21ifx0
    5. h(x)={x21ifx<02ifx0
    6. h(x)={x+2ifx<0(x2)2ifx0
    7. h(x)={(x+10)24ifx<8x+4if8x<4x+4ifx4
    8. f(x)={x+10ifx10|x5|15if10<x2010ifx>20

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

    Part B: Reflections and Dilations

      Match the graph the given function definition.

    1. f(x)=3|x|

    2. f(x)=(x+3)21

    3. f(x)=|x+1|+2

    4. f(x)=x2+1

    5. f(x)=13|x|

    6. f(x)=(x2)2+2

      Use the transformations to graph the following functions.

    1. f(x)=x+5

    2. f(x)=|x|3

    3. g(x)=|x1|

    4. f(x)=(x+2)2

    5. h(x)=x+2

    6. g(x)=x+2

    7. g(x)=(x+2)3

    8. h(x)=x2+1

    9. g(x)=x3+4

    10. f(x)=x2+6

    11. f(x)=3|x|

    12. g(x)=2x2

    13. h(x)=12(x1)2

    14. h(x)=13(x+2)2

    15. g(x)=12x3

    16. f(x)=5x+2

    17. f(x)=4x1+2

    18. h(x)=2x+1

    19. g(x)=14(x+3)31

    20. f(x)=5(x3)2+3

    21. h(x)=3|x+4|2

    22. f(x)=1x

    23. f(x)=1x+2

    24. f(x)=1x+1+2

    Part C: Discussion Board

    1. Use different colors to graph the family of graphs defined by y=kx2, where k{1,12,13,14}. What happens to the graph when the denominator of k is very large? Share your findings on the discussion board.

    2. Graph f(x)=x and g(x)=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.

Answers

  1. e

  2. d

  3. f

  4. y=x; Shift up 3 units; domain: ; range:

  5. y=x2; Shift up 1 unit; domain: ; range: [1,)

  6. y=x2; Shift right 5 units; domain: ; range: [0,)

  7. y=x2; Shift right 5 units and up 2 units; domain: ; range: [2,)

  8. y=|x|; Shift left 4 units; domain: ; range: [0,)

  9. y=|x|; Shift right 1 unit and down 3 units; domain: ; range: [3,)

  10. y=x; Shift down 5 units; domain: [0,); range: [5,)

  11. y=x; Shift right 2 units and up 1 unit; domain: [2,); range: [1,)

  12. y=x3; Shift right 2 units; domain: ; range:

  13. y=x3; Shift right 1 unit and down 4 units; domain: ; range:

  14. y=1x; Shift right 2 units; domain: (,2)(2,); range: (,0)(0,)

  15. y=1x; Shift up 5 units; domain: (,0)(0,); range: (,1)(1,)

  16. y=1x; Shift left 1 unit and down 2 units; domain: (,1)(1,); range: (,2)(2,)

  17. Basic graph y=4; domain: ; range: {−4}

  18. y=x3; Shift up 6 units and right 2 units; domain: ; range:

  19. f(x)=x5

  20. f(x)=(x15)210

  21. f(x)=1x+8+4

  22. f(x)=x+164

  1. b

  2. d

  3. f

  1. Answer may vary

  2. Answer may vary