Section 2.5 Transformations of Functions
Section 2.5 Transformations of Functions Vertical Shifting
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function. (a) g(x) = x2 + 3
(b) h(x) = x2 − 2
EXAMPLE: Use the graph of f (x) = x3 − 9x to sketch the graph of each function. (a) g(x) = x3 − 9x + 10
(b) h(x) = x3 − 9x − 20
1
EXAMPLE: Use the graph of f (x) = x3 − 9x to sketch the graph of each function. (a) g(x) = x3 − 9x + 10
(b) h(x) = x3 − 9x − 20
Horizontal Shifting
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function. (a) g(x) = (x + 4)2
(b) h(x) = (x − 2)2
2
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function. (a) g(x) = (x + 4)2
(b) h(x) = (x − 2)2
EXAMPLE: How is the graph of y = f (x − 3) + 2 obtained from the graph of f ? Answer: The graph shifts right 3 units, then shifts upward 2 units. EXAMPLE: Sketch the graph of f (x) =
√
x − 3 + 4.
Reflecting Graphs
EXAMPLE: Sketch the graph of each function. √ (a) f (x) = −x2 (b) g(x) = −x 3
EXAMPLE: Sketch the graph of each function. √ (a) f (x) = −x2 (b) g(x) = −x
EXAMPLE: Given the graph of f (x) = Step 1: f (x) =
√
√
Step 2: f (x) =
x
√ Step 3: f (x) = − 1 + x (reflection)
EXAMPLE: Given the graph of f (x) =
x, use transformations to graph f (x) = 1 − √
1 + x (horizontal shift)
Step 4: f (x) = 1 −
√
√
1 + x (vertical shift)
x, use transformations to graph f (x) = 1 −
4
√ 1 + x.
√ 1 − x.
EXAMPLE: Given the graph of f (x) = Step 1: f (x) =
√
x, use transformations to graph f (x) = 1 − Step 2: f (x) =
x
√ Step 3: f (x) = − 1 + x (reflection)
Step 5: f (x) = 1 −
√
√
√
1 + x (horizontal shift)
Step 4: f (x) = 1 −
1 − x (reflection about the y-axis)
5
√ 1 − x.
√
1 + x (vertical shift)
Vertical Stretching and Shrinking
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function. 1 (a) g(x) = 3x2 (b) h(x) = x2 3
EXAMPLE: Given the graph of f (x) below, sketch the graph of 12 f (x) + 1.
6
EXAMPLE: Given the graph of f (x) below, sketch the graph of 12 f (x) + 1.
EXAMPLE: Sketch the graph of the function f (x) = 1 − 2(x − 3)2 .
7
Horizontal Stretching and Shrinking
EXAMPLE: The graph of y = f (x) is shown below.
Sketch the graph of each function. (a) y = f (3x)
(b) y = f
1 x 3
EXAMPLE: The graph of y = f (x) is shown below.
Sketch the graph of each function. (a) y = f (2x)
(b) y = f
1 x 2
8
EXAMPLE: The graph of y = f (x) is shown below.
Sketch the graph of each function. (a) y = f (2x)
(b) y = f
1 x 2
Even and Odd Functions
REMARK: Any function is either even, or odd, or neither. PROPERTY: Graphs of even functions are symmetric with respect to the y-axis. Graphs of odd functions are symmetric with respect to the origin. IMPORTANT: Do NOT confuse even/odd functions and even/odd integers!
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EXAMPLES: 1. Functions f (x) = x2 , x4 , x8 , x4 − x2 , x2 + 1, |x|, cos x, etc. are even. In fact, • if f (x) = x2 , then f (−x) = (−x)2 = x2 = f (x) • if f (x) = x4 , then f (−x) = (−x)4 = x4 = f (x) • if f (x) = x8 , then f (−x) = (−x)8 = x8 = f (x) • if f (x) = x4 − x2 , then f (−x) = (−x)4 − (−x)2 = x4 − x2 = f (x) • if f (x) = x2 + 1, then f (−x) = (−x)2 + 1 = x2 + 1 = f (x) • if f (x) = |x|, then f (−x) = | − x| = |x| = f (x) • if f (x) = cos x, then f (−x) = cos(−x) = cos x = f (x) One can see that graphs of all these functions are symmetric with respect to the y-axis. 2. Functions f (x) = x, x3 , x5 , x3 − x7 , sin x, etc. are odd. In fact, • if f (x) = x, then f (−x) = −x = −f (x) • if f (x) = x3 , then f (−x) = (−x)3 = −x3 = −f (x) • if f (x) = x5 , then f (−x) = (−x)5 = −x5 = −f (x) • if f (x) = x3 − x7 , then f (−x) = (−x)3 − (−x)7 = −x3 + x7 = −(x3 − x7 ) = −f (x) • if f (x) = sin x, then f (−x) = sin(−x) = − sin x = −f (x) One can see that graphs of all these functions are symmetric with respect to the origin. 3. Functions f (x) = x + 1, x3 + x2 , x5 − 2, |x − 2| etc. are neither even nor odd. In fact, • if f (x) = x + 1, then f (−1) = −1 + 1 = 0,
f (1) = 1 + 1 = 2
so f (−1) 6= ±f (1). Therefore f (x) = x + 1 is neither even nor odd. • if f (x) = x3 + x2 , then f (−1) = (−1)3 + (−1)2 = −1 + 1 = 0,
f (1) = 13 + 12 = 2
so f (−1) 6= ±f (1). Therefore f (x) = x3 + x2 is neither even nor odd. • if f (x) = x5 − 2, then f (−1) = (−1)5 − 2 = −1 − 2 = −3,
f (1) = 15 − 2 = 1 − 2 = −1
so f (−1) 6= ±f (1). Therefore f (x) = x5 − 2 is neither even nor odd. • if f (x) = |x − 2|, then f (−1) = | − 1 − 2| = | − 3| = 3,
f (1) = |1 − 2| = | − 1| = 1
so f (−1) 6= ±f (1). Therefore f (x) = |x − 2| is neither even nor odd.
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EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function. (a) g(x) = x2 + 3
(b) h(x) = x2 − 2
EXAMPLE: Use the graph of f (x) = x3 − 9x to sketch the graph of each function. (a) g(x) = x3 − 9x + 10
(b) h(x) = x3 − 9x − 20
1
EXAMPLE: Use the graph of f (x) = x3 − 9x to sketch the graph of each function. (a) g(x) = x3 − 9x + 10
(b) h(x) = x3 − 9x − 20
Horizontal Shifting
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function. (a) g(x) = (x + 4)2
(b) h(x) = (x − 2)2
2
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function. (a) g(x) = (x + 4)2
(b) h(x) = (x − 2)2
EXAMPLE: How is the graph of y = f (x − 3) + 2 obtained from the graph of f ? Answer: The graph shifts right 3 units, then shifts upward 2 units. EXAMPLE: Sketch the graph of f (x) =
√
x − 3 + 4.
Reflecting Graphs
EXAMPLE: Sketch the graph of each function. √ (a) f (x) = −x2 (b) g(x) = −x 3
EXAMPLE: Sketch the graph of each function. √ (a) f (x) = −x2 (b) g(x) = −x
EXAMPLE: Given the graph of f (x) = Step 1: f (x) =
√
√
Step 2: f (x) =
x
√ Step 3: f (x) = − 1 + x (reflection)
EXAMPLE: Given the graph of f (x) =
x, use transformations to graph f (x) = 1 − √
1 + x (horizontal shift)
Step 4: f (x) = 1 −
√
√
1 + x (vertical shift)
x, use transformations to graph f (x) = 1 −
4
√ 1 + x.
√ 1 − x.
EXAMPLE: Given the graph of f (x) = Step 1: f (x) =
√
x, use transformations to graph f (x) = 1 − Step 2: f (x) =
x
√ Step 3: f (x) = − 1 + x (reflection)
Step 5: f (x) = 1 −
√
√
√
1 + x (horizontal shift)
Step 4: f (x) = 1 −
1 − x (reflection about the y-axis)
5
√ 1 − x.
√
1 + x (vertical shift)
Vertical Stretching and Shrinking
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function. 1 (a) g(x) = 3x2 (b) h(x) = x2 3
EXAMPLE: Given the graph of f (x) below, sketch the graph of 12 f (x) + 1.
6
EXAMPLE: Given the graph of f (x) below, sketch the graph of 12 f (x) + 1.
EXAMPLE: Sketch the graph of the function f (x) = 1 − 2(x − 3)2 .
7
Horizontal Stretching and Shrinking
EXAMPLE: The graph of y = f (x) is shown below.
Sketch the graph of each function. (a) y = f (3x)
(b) y = f
1 x 3
EXAMPLE: The graph of y = f (x) is shown below.
Sketch the graph of each function. (a) y = f (2x)
(b) y = f
1 x 2
8
EXAMPLE: The graph of y = f (x) is shown below.
Sketch the graph of each function. (a) y = f (2x)
(b) y = f
1 x 2
Even and Odd Functions
REMARK: Any function is either even, or odd, or neither. PROPERTY: Graphs of even functions are symmetric with respect to the y-axis. Graphs of odd functions are symmetric with respect to the origin. IMPORTANT: Do NOT confuse even/odd functions and even/odd integers!
9
EXAMPLES: 1. Functions f (x) = x2 , x4 , x8 , x4 − x2 , x2 + 1, |x|, cos x, etc. are even. In fact, • if f (x) = x2 , then f (−x) = (−x)2 = x2 = f (x) • if f (x) = x4 , then f (−x) = (−x)4 = x4 = f (x) • if f (x) = x8 , then f (−x) = (−x)8 = x8 = f (x) • if f (x) = x4 − x2 , then f (−x) = (−x)4 − (−x)2 = x4 − x2 = f (x) • if f (x) = x2 + 1, then f (−x) = (−x)2 + 1 = x2 + 1 = f (x) • if f (x) = |x|, then f (−x) = | − x| = |x| = f (x) • if f (x) = cos x, then f (−x) = cos(−x) = cos x = f (x) One can see that graphs of all these functions are symmetric with respect to the y-axis. 2. Functions f (x) = x, x3 , x5 , x3 − x7 , sin x, etc. are odd. In fact, • if f (x) = x, then f (−x) = −x = −f (x) • if f (x) = x3 , then f (−x) = (−x)3 = −x3 = −f (x) • if f (x) = x5 , then f (−x) = (−x)5 = −x5 = −f (x) • if f (x) = x3 − x7 , then f (−x) = (−x)3 − (−x)7 = −x3 + x7 = −(x3 − x7 ) = −f (x) • if f (x) = sin x, then f (−x) = sin(−x) = − sin x = −f (x) One can see that graphs of all these functions are symmetric with respect to the origin. 3. Functions f (x) = x + 1, x3 + x2 , x5 − 2, |x − 2| etc. are neither even nor odd. In fact, • if f (x) = x + 1, then f (−1) = −1 + 1 = 0,
f (1) = 1 + 1 = 2
so f (−1) 6= ±f (1). Therefore f (x) = x + 1 is neither even nor odd. • if f (x) = x3 + x2 , then f (−1) = (−1)3 + (−1)2 = −1 + 1 = 0,
f (1) = 13 + 12 = 2
so f (−1) 6= ±f (1). Therefore f (x) = x3 + x2 is neither even nor odd. • if f (x) = x5 − 2, then f (−1) = (−1)5 − 2 = −1 − 2 = −3,
f (1) = 15 − 2 = 1 − 2 = −1
so f (−1) 6= ±f (1). Therefore f (x) = x5 − 2 is neither even nor odd. • if f (x) = |x − 2|, then f (−1) = | − 1 − 2| = | − 3| = 3,
f (1) = |1 − 2| = | − 1| = 1
so f (−1) 6= ±f (1). Therefore f (x) = |x − 2| is neither even nor odd.
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