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2.3E: Exercises - Using Transformations to Graph Functions

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    Exercise \(\PageIndex{3}\)

    Match the graph to the function definition.

    caf84d0d27db512ef90d11b59b6c37dc.png
    Figure 2.5.18
    2fe54b1c80ea84f0f721462f90455c0b.png
    Figure 2.5.19
    3622a0d2256166544a122ecd7156de36.png
    Figure 2.5.20
    47088a9efd6814511cb0fc8d233b539f.png
    Figure 2.5.21
    d44d62205d34ed371aad179b77c54a81.png
    Figure 2.5.22
    Figure 2.5.23
    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. \(f(x) = |x + 2| − 2\)
    Answer

    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. \(f ( x ) = \sqrt [ 3 ] { x + 8 } - 4\)
    Answer

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

    ed14f13811bfb7c397b768ab1e6d718a.png
    Figure 2.5.24

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

    0e393f0d6e151259a123b1e505dec86b.png
    Figure 2.5.25

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

    8fc7f879a8ba5f12d0b98f348e5adadb.png
    Figure 2.5.26

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

    57a5fd7bcf0e225b10961c6534cd4545.png
    Figure 2.5.27

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

    86a1d10b4aad0ab79bc2c8dd55bf4f38.png
    Figure 2.5.28

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

    424b66df0df22a96fd88c4957413d44e.png
    Figure 2.5.29

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

    457665c1ea5709240bd4c6e1685a1985.png
    Figure 2.5.30

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

    da6d3f21b303aeb0b29fe4975b48a64f.png
    Figure 2.5.31

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

    01b74b05906d95ff14c5aa6de0ae7b4f.png
    Figure 2.5.32

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

    a4f584febcd95dc5ef92bbe2ef80df7c.png
    Figure 2.5.33

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

    75fa23d883d738eeb47a020057002b8f.png
    Figure 2.5.34

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

    53d3a12d61be06d8913ae13668760ebb.png
    Figure 2.5.35

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

    0eac4ad67881e57bfa8e7dc46c933e8e.png
    Figure 2.5.36

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

    dec428893d68980da985eabaf7f7fb11.png
    Figure 2.5.37

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

    43dafc2ae310a7b8dbba8ee467325ad8.png
    Figure 2.5.38

    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. \(f ( x ) = \left\{ \begin{array} { l l } { x + 10 } & { \text { if } x \leq - 10 } \\ { | x - 5 | - 15 } & { \text { if } - 10 < x \leq 20 } \\ { 10 } & { \text { if } x > 20 } \end{array} \right.\)
    Answer

    1.

    1790a36f5e4c391f1d37b3abdabb2349.png
    Figure 2.5.39

    3.

    f2e8945e9fea8dc040b6d5a1180fd1d0.png
    Figure 2.5.40

    5.

    ab01cd028abe7241da8e857be88bdb8a.png
    Figure 2.5.41

    7.

    00235242b3f9ae8ded77603b43125c75.png
    Figure 2.5.42

    Exercise \(\PageIndex{6}\)

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

    1.

    9de0ccbee5d59fa05acab85d085cb4d7.png
    Figure 2.5.43

    2.

    122abd4ccb8eb72a59532b22ed6116ab.png
    Figure 2.5.44

    3.

    b8f5c01476fe7ee9f21dd781da420d2d.png
    Figure 2.5.45

    4.

    Figure 2.5.46

    5.

    e90ad3255312e9bf25d0f866de703eb4.png
    Figure 2.5.47

    6.

    6c160b69a9ef56763a5424ea14fbc86f.png
    Figure 2.5.48

    7.

    613f12af91bfbf853201387cb6dd7acb.png
    Figure 2.5.49

    8.

    Figure 2.5.50
    Answer

    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.

    19f3c208cfdeeffde7e76281b4b28f46.png
    Figure 2.5.51
    7ddabfc77a72214e9f6bea00e3b2cca0.png
    Figure 2.5.52
    039e6f4a86d07a578660882bccf7ea40.png
    Figure 2.5.53
    16b19343fd01aecf51c1cdea8af3ee21.png
    Figure 2.5.54
    26cdff42b4eb188a4512c934fd59f9e5.png
    Figure 2.5.55
    75295519ff6aaa13dced0dc6ed6e2ef7.png
    Figure 2.5.56
    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. \(f ( x ) = - ( x - 2 ) ^ { 2 } + 2\)
    Answer

    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. \(f ( x ) = - \frac { 1 } { x + 1 } + 2\)
    Answer

    1.

    7ab442c285558a001bd71c319dfceb86.png
    Figure 2.5.57

    3.

    8e5290466d22bfaad7a33f4ffcc1c2d0.png
    Figure 2.5.58

    5.

    73fbbd2cc539ff2ab99df22497167aec.png
    Figure 2.5.59

    7.

    87909d16e900cb252e880491550fd960.png
    Figure 2.5.60

    9.

    92bf8584935a01fd897e3af4c08fa4fd.png
    Figure 2.5.61

    11.

    b03918836eb8805c137b8a53dc8d07ff.png
    Figure 2.5.62

    13.

    ba1195d282dfeb1b813eeec2a0ff6e74.png
    Figure 2.5.63

    15.

    576a64916f5e6c8c4b851efadf07189d.png
    Figure 2.5.64

    17.

    3a6abc9abd61596a77ecfe672f89976d.png
    Figure 2.5.65

    19.

    Figure 2.5.66

    21.

    Figure 2.5.67

    23.

    Figure 2.5.68

    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.
    Answer

    1. Answer may vary

    3. Answer may vary

    2.3E: Exercises - Using Transformations to Graph Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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