Stewart Precalc 6e Section 2.5: Transformations of Functions

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Stewart Precalc 6e Section 2.5: Transformations of Functions 0/101
1 Concepts - Vertical and Horizontal Shifts · Level 1
Fill in the blank with the appropriate direction (left, right, up, or down).
(a) The graph of \(y = f(x) + 3\) is obtained from the graph of \(y = f(x)\) by shifting _____ 3 units.
(b) The graph of \(y = f(x + 3)\) is obtained from the graph of \(y = f(x)\) by shifting _____ 3 units.

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2 Concepts - Vertical and Horizontal Shifts · Level 1
Fill in the blank with the appropriate direction (left, right, up, or down).
(a) The graph of \(y = f(x) - 3\) is obtained from the graph of \(y = f(x)\) by shifting _____ 3 units.
(b) The graph of \(y = f(x - 3)\) is obtained from the graph of \(y = f(x)\) by shifting _____ 3 units.

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3 Concepts - Reflections · Level 1
Fill in the blank with the appropriate axis (\(x\)-axis or \(y\)-axis).
(a) The graph of \(y = -f(x)\) is obtained from the graph of \(y = f(x)\) by reflecting in the _____.
(b) The graph of \(y = f(-x)\) is obtained from the graph of \(y = f(x)\) by reflecting in the _____.

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4 Concepts - Match Transformations · Level 2
Match the graph with the function.
question image
(a) \(y = |x + 1|\)
(b) \(y = |x - 1|\)
(c) \(y = |x| - 1\)
(d) \(y = -|x|\)

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5 Skills - Describe Transformations · Level 1
(a) \(y = f(x) - 5\)
(b) \(y = f(x - 5)\)

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6 Skills - Describe Transformations · Level 1
(a) \(y = f(x + 7)\)
(b) \(y = f(x) + 7\)

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7 Skills - Describe Transformations · Level 1
(a) \(y = -f(x)\)
(b) \(y = f(-x)\)

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8 Skills - Describe Transformations · Level 2
(a) \(y = -2 f(x)\)
(b) \(y = -\dfrac{1}{2} f(x)\)

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9 Skills - Describe Transformations · Level 2
(a) \(y = -f(x) + 5\)
(b) \(y = 3 f(x) - 5\)

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10 Skills - Describe Transformations · Level 2
(a) \(y = f(x - 4) + \dfrac{3}{4}\)
(b) \(y = f(x + 4) - \dfrac{3}{4}\)

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11 Skills - Describe Transformations · Level 2
(a) \(y = 2 f(x + 1) - 3\)
(b) \(y = 2 f(x - 1) + 3\)

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12 Skills - Describe Transformations · Level 2
(a) \(y = 3 - 2 f(x)\)
(b) \(y = 2 - f(-x)\)

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13 Skills - Describe Transformations · Level 2
(a) \(y = f(4 x)\)
(b) \(y = f\left(\dfrac{1}{4} x\right)\)

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14 Skills - Describe Transformations · Level 2
(a) \(y = f(2 x) - 1\)
(b) \(y = 2 f\left(\dfrac{1}{2} x\right)\)

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15 Skills - Explain Transformations · Level 1
Explain how the graph of \(g\) is obtained from the graph of \(f\).
(a) \(f(x) = x^2\), \(g(x) = (x + 2)^2\)
(b) \(f(x) = x^2\), \(g(x) = x^2 + 2\)

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16 Skills - Explain Transformations · Level 1
Explain how the graph of \(g\) is obtained from the graph of \(f\).
(a) \(f(x) = x^3\), \(g(x) = (x - 4)^3\)
(b) \(f(x) = x^3\), \(g(x) = x^3 - 4\)

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17 Skills - Explain Transformations · Level 2
Explain how the graph of \(g\) is obtained from the graph of \(f\).
(a) \(f(x) = |x|\), \(g(x) = |x + 2| - 2\)
(b) \(f(x) = |x|\), \(g(x) = |x - 2| + 2\)

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18 Skills - Explain Transformations · Level 2
Explain how the graph of \(g\) is obtained from the graph of \(f\).
(a) \(f(x) = \sqrt{x}\), \(g(x) = -\sqrt{x} + 1\)
(b) \(f(x) = \sqrt{x}\), \(g(x) = \sqrt{-x} + 1\)

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19 Skills - Graph from Standard · Level 2
Use the graph of \(y = x^2\) in Figure 4 to graph the following.
(a) \(g(x) = x^2 + 1\)
(b) \(g(x) = (x - 1)^2\)
(c) \(g(x) = -x^2\)
(d) \(g(x) = (x - 1)^2 + 3\)

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20 Skills - Graph from Standard · Level 2
Use the graph of \(y = \sqrt{x}\) in Figure 5 to graph the following.
(a) \(g(x) = \sqrt{x - 2}\)
(b) \(g(x) = \sqrt{x} + 1\)
(c) \(g(x) = \sqrt{x + 2} + 2\)
(d) \(g(x) = -\sqrt{x} + 1\)

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21 Skills - Sketch by Transformations · Level 2
\( f(x) = x^2 - 1 \)
22 Skills - Sketch by Transformations · Level 2
\( f(x) = x^2 + 5 \)
23 Skills - Sketch by Transformations · Level 2
\( f(x) = \sqrt{x} + 1 \)
24 Skills - Sketch by Transformations · Level 2
\( f(x) = |x| - 1 \)
25 Skills - Sketch by Transformations · Level 2
\( f(x) = (x - 5)^2 \)
26 Skills - Sketch by Transformations · Level 2
\( f(x) = (x + 1)^2 \)
27 Skills - Sketch by Transformations · Level 2
\( f(x) = \sqrt{x + 4} \)
28 Skills - Sketch by Transformations · Level 2
\( f(x) = |x - 3| \)
29 Skills - Sketch by Transformations · Level 2
\( f(x) = -x^3 \)
30 Skills - Sketch by Transformations · Level 2
\( f(x) = -|x| \)
31 Skills - Sketch by Transformations · Level 2
\( y = \sqrt[4]{-x} \)
32 Skills - Sketch by Transformations · Level 2
\( y = \sqrt[3]{-x} \)
33 Skills - Sketch by Transformations · Level 2
\( y = \dfrac{1}{4} x^2 \)
34 Skills - Sketch by Transformations · Level 2
\( y = -5 \sqrt{x} \)
35 Skills - Sketch by Transformations · Level 2
\( y = 3 |x| \)
36 Skills - Sketch by Transformations · Level 2
\( y = \dfrac{1}{2} |x| \)
37 Skills - Sketch by Transformations · Level 3
\( y = (x - 3)^2 + 5 \)
38 Skills - Sketch by Transformations · Level 3
\( y = \sqrt{x + 4} - 3 \)
39 Skills - Sketch by Transformations · Level 3
\( y = 3 - \dfrac{1}{2}(x - 1)^2 \)
40 Skills - Sketch by Transformations · Level 3
\( y = 2 - \sqrt{x + 1} \)
41 Skills - Sketch by Transformations · Level 3
\( y = |x + 2| + 2 \)
42 Skills - Sketch by Transformations · Level 3
\( y = 2 - |x| \)
43 Skills - Sketch by Transformations · Level 3
\( y = \dfrac{1}{2} \sqrt{x + 4} - 3 \)
44 Skills - Sketch by Transformations · Level 3
\( y = 3 - 2(x - 1)^2 \)
45 Skills - Write Transformed Equation · Level 1
\(f(x) = x^2\); shift upward 3 units
46 Skills - Write Transformed Equation · Level 1
\(f(x) = x^3\); shift downward 1 unit
47 Skills - Write Transformed Equation · Level 1
\(f(x) = \sqrt{x}\); shift 2 units to the left
48 Skills - Write Transformed Equation · Level 1
\(f(x) = \sqrt[3]{x}\); shift 1 unit to the right
49 Skills - Write Transformed Equation · Level 2
\(f(x) = |x|\); shift 3 units to the right and shift upward 1 unit
50 Skills - Write Transformed Equation · Level 2
\(f(x) = |x|\); shift 4 units to the left and shift downward 2 units
51 Skills - Write Transformed Equation · Level 2
\(f(x) = \sqrt[4]{x}\); reflect in the \(y\)-axis and shift upward 1 unit
52 Skills - Write Transformed Equation · Level 2
\(f(x) = x^2\); shift 2 units to the left and reflect in the \(x\)-axis
53 Skills - Write Transformed Equation · Level 3
\(f(x) = x^2\); stretch vertically by a factor of 2, shift downward 2 units, and shift 3 units to the right
54 Skills - Write Transformed Equation · Level 3
\(f(x) = |x|\); shrink vertically by a factor of \(\dfrac{1}{2}\), shift to the left 1 unit, and shift upward 3 units
55 Skills - Find Formula from Graphs · Level 3
The graphs of \(f\) and \(g\) are given. Find a formula for the function \(g\).
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56 Skills - Find Formula from Graphs · Level 3
The graphs of \(f\) and \(g\) are given. Find a formula for the function \(g\).
question image
57 Skills - Find Formula from Graphs · Level 3
The graphs of \(f\) and \(g\) are given. Find a formula for the function \(g\).
question image
58 Skills - Find Formula from Graphs · Level 3
The graphs of \(f\) and \(g\) are given. Find a formula for the function \(g\).
question image
59 Skills - Find Formula from Graphs · Level 3
The graphs of \(f\) and \(g\) are given. Find a formula for the function \(g\).
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60 Skills - Find Formula from Graphs · Level 3
The graphs of \(f\) and \(g\) are given. Find a formula for the function \(g\).
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61 Skills - Match Equations to Graphs · Level 2
The graph of \(y = f(x)\) is given. Match each equation with its graph.
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(a) \(y = f(x - 4)\)
(b) \(y = f(x) + 3\)
(c) \(y = 2 f(x + 6)\)
(d) \(y = -f(2 x)\)

Enter your answer directly below each part above.

62 Skills - Match Equations to Graphs · Level 2
The graph of \(y = f(x)\) is given. Match each equation with its graph.
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(a) \(y = \dfrac{1}{3} f(x)\)
(b) \(y = -f(x + 4)\)
(c) \(y = f(x - 4) + 3\)

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63 Skills - Sketch from Given Graph · Level 3
The graph of \(f\) is given. Sketch the graphs of the following functions.
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(a) \(y = f(x - 2)\)
(b) \(y = f(x) - 2\)
(c) \(y = 2 f(x)\)
(d) \(y = -f(x) + 3\)
(e) \(y = f(-x)\)
(f) \(y = \dfrac{1}{2} f(x - 1)\)

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64 Skills - Sketch from Given Graph · Level 3
The graph of \(g\) is given. Sketch the graphs of the following functions.
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(a) \(y = g(x + 1)\)
(b) \(y = g(-x)\)
(c) \(y = g(x - 2)\)
(d) \(y = g(x) - 2\)
(e) \(y = -g(x)\)
(f) \(y = 2 g(x)\)

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65 Skills - Horizontal Stretches and Shrinks · Level 3
The graph of \(g\) is given. Use it to graph each of the following functions.
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(a) \(y = g(2 x)\)
(b) \(y = g\left(\dfrac{1}{2} x\right)\)

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66 Skills - Horizontal Stretches and Shrinks · Level 3
The graph of \(h\) is given. Use it to graph each of the following functions.
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(a) \(y = h(3 x)\)
(b) \(y = h\left(\dfrac{1}{3} x\right)\)

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67 Skills - Floor Function Transformation · Level 3
Use the graph of \(f(x) = \lfloor x \rfloor\) described on page 156 to graph the indicated function. \(y = \lfloor 2 x \rfloor\)
68 Skills - Floor Function Transformation · Level 3
Use the graph of \(f(x) = \lfloor x \rfloor\) described on page 156 to graph the indicated function. \(y = \lfloor \dfrac{1}{4} x \rfloor\)
69 Skills - Graphing Calculator · Level 2
Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-8, 8]\) by \([-2, 8]\)
(a) \(y = \sqrt[4]{x}\)
(b) \(y = \sqrt[4]{x + 5}\)
(c) \(y = 2 \sqrt[4]{x + 5}\)
(d) \(y = 4 + 2 \sqrt[4]{x + 5}\)

Enter your answer directly below each part above.

70 Skills - Graphing Calculator · Level 2
Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-8, 8]\) by \([-6, 6]\)
(a) \(y = |x|\)
(b) \(y = -|x|\)
(c) \(y = -3 |x|\)
(d) \(y = -3 |x - 5|\)

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71 Skills - Graphing Calculator · Level 2
Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-4, 6]\) by \([-4, 4]\)
(a) \(y = x^6\)
(b) \(y = \dfrac{1}{3} x^6\)
(c) \(y = -\dfrac{1}{3} x^6\)
(d) \(y = -\dfrac{1}{3} (x - 4)^6\)

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72 Skills - Graphing Calculator · Level 2
Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-6, 6]\) by \([-4, 4]\)
(a) \(y = \dfrac{1}{\sqrt{x}}\)
(b) \(y = \dfrac{1}{\sqrt{x + 3}}\)
(c) \(y = \dfrac{1}{2 \sqrt{x + 3}}\)
(d) \(y = \dfrac{1}{2 \sqrt{x + 3}} - 3\)

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73 Skills - Calculator with Horizontal Scaling · Level 3
If \(f(x) = \sqrt{2 x - x^2}\), graph the following functions in the viewing rectangle \([-5, 5]\) by \([-4, 4]\). How is each graph related to the graph in part (a)?
(a) \(y = f(x)\)
(b) \(y = f(2 x)\)
(c) \(y = f\left(\dfrac{1}{2} x\right)\)

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74 Skills - Calculator with Reflections and Scaling · Level 3
If \(f(x) = \sqrt{2 x - x^2}\), graph the following functions in the viewing rectangle \([-5, 5]\) by \([-4, 4]\). How is each graph related to the graph in part (a)?
(a) \(y = f(x)\)
(b) \(y = f(-x)\)
(c) \(y = -f(-x)\)
(d) \(y = f(-2 x)\)
(e) \(y = f\left(-\dfrac{1}{2} x\right)\)

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75 Skills - Even/Odd Functions · Level 2
Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. \(f(x) = x^4\)
76 Skills - Even/Odd Functions · Level 2
Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. \(f(x) = x^3\)
77 Skills - Even/Odd Functions · Level 2
Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. \(f(x) = x^2 + x\)
78 Skills - Even/Odd Functions · Level 2
Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. \(f(x) = x^4 - 4 x^2\)
79 Skills - Even/Odd Functions · Level 2
Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. \(f(x) = x^3 - x\)
80 Skills - Even/Odd Functions · Level 2
Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. \(f(x) = 3 x^3 + 2 x^2 + 1\)
81 Skills - Even/Odd Functions · Level 2
Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. \(f(x) = 1 - \sqrt[3]{x}\)
82 Skills - Even/Odd Functions · Level 2
Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. \(f(x) = x + \dfrac{1}{x}\)
83 Skills - Complete Graph for Even/Odd · Level 2
The graph of a function defined for \(x \geq 0\) is given. Complete the graph for \(x < 0\) to make (a) an even function and (b) an odd function.
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84 Skills - Complete Graph for Even/Odd · Level 2
The graph of a function defined for \(x \geq 0\) is given. Complete the graph for \(x < 0\) to make (a) an even function and (b) an odd function.
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85 Skills - Absolute Value of Function · Level 3
These exercises show how the graph of \(y = |f(x)|\) is obtained from the graph of \(y = f(x)\). The graphs of \(f(x) = x^2 - 4\) and \(g(x) = |x^2 - 4|\) are shown. Explain how the graph of \(g\) is obtained from the graph of \(f\).
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86 Skills - Absolute Value of Function · Level 3
The graph of \(f(x) = x^4 - 4 x^2\) is shown. Use this graph to sketch the graph of \(g(x) = |x^4 - 4 x^2|\).
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87 Skills - Sketch Function and Absolute Value · Level 3
Sketch the graph of each function.
(a) \(f(x) = 4 x - x^2\)
(b) \(g(x) = |4 x - x^2|\)

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88 Skills - Sketch Function and Absolute Value · Level 3
Sketch the graph of each function.
(a) \(f(x) = x^3\)
(b) \(g(x) = |x^3|\)

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89 Applications - Sales Growth · Level 3
Sales Growth The annual sales of a certain company can be modeled by the function \(f(t) = 4 + 0.01 t^2\), where \(t\) represents years since 1990 and \(f(t)\) is measured in millions of dollars.
(a) What shifting and shrinking operations must be performed on the function \(y = t^2\) to obtain the function \(y = f(t)\)?
(b) Suppose you want \(t\) to represent years since 2000 instead of 1990. What transformation would you have to apply to the function \(y = f(t)\) to accomplish this? Write the new function \(y = g(t)\) that results from this transformation.

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90 Applications - Temperature Scale · Level 3
Changing Temperature Scales The temperature on a certain afternoon is modeled by the function \(C(t) = \dfrac{1}{2} t^2 + 2\) where \(t\) represents hours after 12 noon \((0 \leq t \leq 6)\) and \(C\) is measured in \(^{\circ}\)C.
(a) What shifting and shrinking operations must be performed on the function \(y = t^2\) to obtain the function \(y = C(t)\)?
(b) Suppose you want to measure the temperature in \(^{\circ}\)F instead. What transformation would you have to apply to the function \(y = C(t)\) to accomplish this? (Use the fact that the relationship between Celsius and Fahrenheit degrees is given by \(F = \dfrac{9}{5} C + 32\).) Write the new function \(y = F(t)\) that results from this transformation.

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91 Discovery - Sums of Even and Odd Functions · Level 3
Sums of Even and Odd Functions If \(f\) and \(g\) are both even functions, is \(f + g\) necessarily even? If both are odd, is their sum necessarily odd? What can you say about the sum if one is odd and one is even? In each case, prove your answer.
92 Discovery - Products of Even and Odd Functions · Level 3
Products of Even and Odd Functions Answer the same questions as in Exercise 91, except this time consider the product of \(f\) and \(g\) instead of the sum.
93 Discovery - Even and Odd Power Functions · Level 2
Even and Odd Power Functions What must be true about the integer \(n\) if the function \(f(x) = x^n\) is an even function? If it is an odd function? Why do you think the names "even" and "odd" were chosen for these function properties?
94 Example - Vertical Shifts of Graphs · Level 1
Use the graph of \(f(x) = x^2\) to sketch the graph of each function.
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(a) \(g(x) = x^2 + 3\)
(b) \(h(x) = x^2 - 2\)

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95 Example - Horizontal Shifts of Graphs · Level 2
Use the graph of \(f(x) = x^2\) to sketch the graph of each function.
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(a) \(g(x) = (x + 4)^2\)
(b) \(h(x) = (x - 2)^2\)

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96 Example - Combining Horizontal and Vertical Shifts · Level 2
Sketch the graph of \(f(x) = \sqrt{x - 3} + 4\).
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97 Example - Reflecting Graphs · Level 2
Sketch the graph of each function.
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(a) \(f(x) = -x^2\)
(b) \(g(x) = \sqrt{-x}\)

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98 Example - Vertical Stretching and Shrinking of Graphs · Level 2
Use the graph of \(f(x) = x^2\) to sketch the graph of each function.
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(a) \(g(x) = 3 x^2\)
(b) \(h(x) = \dfrac{1}{3} x^2\)

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99 Example - Combining Shifting, Stretching, and Reflecting · Level 3
Sketch the graph of the function \(f(x) = 1 - 2(x - 3)^2\).
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100 Example - Horizontal Stretching and Shrinking of Graphs · Level 3
The graph of \(y = f(x)\) is shown in Figure 8. Sketch the graph of each function.
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(a) \(y = f(2 x)\)
(b) \(y = f\left(\dfrac{1}{2} x\right)\)

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101 Example - Even and Odd Functions · Level 2
Determine whether the functions are even, odd, or neither even nor odd.
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(a) \(f(x) = x^5 + x\)
(b) \(g(x) = 1 - x^4\)
(c) \(h(x) = 2x - x^2\)

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Answered: 0 / 101