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Published byKristopher Sullens Modified over 2 years ago

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1 To find the x-intercepts of y = f (x), set y = 0 and solve for x. INTERCEPTS AND ZEROS To find the y-intercepts of y = f (x), set x = 0; the y-intercept is f (0). x-intercepts correspond to the zeros of the function f(x) = x 2 – x – 2 x 2 – x – 2 = 0 (x + 1)(x – 2) = 0 x = – 1 f(0) = (0) 2 – 0 – 2 = – 2 ( – 1, 0) (0, – 2) (2, 0) x = 2

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2 EXAMPLE 1 f(x) = 3x 2 + 7x – 6 x-intercepts (Set y = 0) 3x 2 + 7x – 6 = 0 3x 2 + 9x – 2x – 6 = 0 3x (x + 3) – 2(x + 3) = 0 (3x – 2)(x + 3) = 0 x = 2 / 3 y-intercept (Set x = 0) f(0) = 3(0) 2 + 7(0) – 6 f(0) = – 6 ( – 3, 0) (0, -6) ( 2 /3, 0) x = -3

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3 EXAMPLE 2 f(x) = 2 sin x, - < x < x-intercepts (Set y = 0)y-intercept (Set x = 0) (0, 1) (0, – 1) y 1 (1, 0) x (– 1, 0)

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4 EXAMPLE 3 f(x) = 2cos x – 1, 0 < x < 2 x-intercepts (Set y = 0)y-intercept (Set x = 0) 1 2 2

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5 Symmetry An even function satisfies f (– x) = f ( x ). The graph of an even function is symmetric about the y-axis ( x, y)( – x, y)

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6 EXAMPLE 4 f(x) = 9 – x 2 f(– x) = 9 – (– x) 2 f(– x) = 9 – x 2 Show that this is an even function. It is symmetrical to the y-axis (2, 5)( – 2, 5) ( – 3, 0)(3, 0)

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7 EXAMPLE 5 f(x) = x 4 – 4x 2 f(– x) = (–x) 4 – 4(– x) 2 f(– x) = x 4 – 4x 2 Show that this is an even function. It is symmetrical to the y-axis ( – 2, 0)(2, 0) (1, – 3) ( – 1, – 3)

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8 The graph of an odd function is symmetric about the origin. An odd function satisfies f (– x) = – f ( x ) (x, y) (-x, -y)

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9 EXAMPLE 6 f(x) = x 3 – 9x Show that this is an odd function. It is symmetrical to the origin – f(x) = – (x 3 – 9x ) – f(x) = – x 3 + 9x An odd function satisfies f (– x) = – f ( x ) (– 2, 10) (2, – 10) f(-x) = (-x) 3 – 9(-x) f(-x) = – x 3 + 9x

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10 EXAMPLE 7 Determine if the function is even, odd, or neither. Find f(-x): Since f(x) ≠ f( – x), the function is not even and not symmetric about the y-axis. Find –f(x): Since f( – x) = – f(x) the function is odd and symmetric about the origin =

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11 EXAMPLE 8 y = 4x 2 – x Determine if the function is even, odd, or neither. Since f(x) ≠ f( – x), the function is not even and not symmetric about the y-axis. Since f( – x) ≠ – f(x) the function is not odd and not symmetric about the origin Find f( – x): Find –f(x): ≠

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