Unit 6 Lesson #1 Intercepts and Symmetry INTERCEPTS AND ZEROS To find the x-intercepts of y = f (x), set y = 0 and solve for x. x-intercepts correspond to the zeros of the function f(x) = x2 – x – 2 x2 – x – 2 = 0 (x + 1)(x – 2) = 0 (– 1, 0) (2, 0) x = –1 x = 2 To find the y-intercepts of y = f (x), set x = 0; the y-intercept is f (0). (0, – 2) f(0) = (0)2 – 0 – 2 = – 2

Unit 6 Lesson #1 Intercepts and Symmetry EXAMPLE 1 f(x) = 3x2 + 7x – 6 x-intercepts (Set y = 0) y-intercept (Set x = 0) 3x2 + 7x – 6 = 0 f(0) = 3(0)2 + 7(0) – 6 3x2 + 9x – 2x – 6 = 0 f(0) = – 6 3x (x + 3) – 2(x + 3) = 0 (3x – 2)(x + 3) = 0 (– 3, 0) (2/3, 0) x = 2/3 x = -3 (0, -6)

EXAMPLE 2 f(x) = 2 sin x, -p < x < p x-intercepts (Set y = 0) y-intercept (Set x = 0) 𝟐 𝐬𝐢𝐧 𝒙=𝟎 𝐬𝐢𝐧 𝒙=𝟎 𝒙=𝐬𝐢𝐧 −𝟏 𝟎=𝟎   𝒇 𝟎 =𝟐 𝐬𝐢𝐧 𝟎   𝒇 𝟎 =𝟐 𝟎 =𝟎 Also 𝐬𝐢𝐧 (𝝅)=𝟎 𝐬𝐢𝐧 (−𝝅)=𝟎   q (0, 1) (0, – 1) y 1 (1, 0) x (– 1, 0)

EXAMPLE 3 f(x) = 2cos x – 1, 0 < x < 2p x-intercepts (Set y = 0) y-intercept (Set x = 0) 𝟐 𝐜𝐨𝐬 𝒙−𝟏=𝟎 𝐜𝐨𝐬 𝒙= 𝟏 𝟐 𝒇 𝟎 =𝟐 𝐜𝐨𝐬 𝟎−𝟏 𝒇 𝟎 =𝟐 𝟏 −𝟏 𝒇 𝟎 =𝟏   1 2 𝒙=𝐜𝐨𝐬 −𝟏 𝟏 𝟐 =𝟔𝟎° = 𝝅 𝟑   𝒙=𝟑𝟎𝟎°= 𝟓𝝅 𝟑

Unit 6 Lesson #1 Intercepts and Symmetry An even function satisfies f (– x) = f ( x ). ( x , y) (– x , y) The graph of an even function is symmetric about the y-axis

Unit 6 Lesson #1 Intercepts and Symmetry EXAMPLE 4 f(x) = 9 – x2 Show that this is an even function. It is symmetrical to the y-axis f(– x) = 9 – (– x)2 f(– x) = 9 – x2 (– 2, 5) (2, 5) (– 3, 0) (3, 0)

Unit 6 Lesson #1 Intercepts and Symmetry EXAMPLE 5 f(x) = x4 – 4x2 Show that this is an even function. It is symmetrical to the y-axis f(– x) = (–x)4 – 4(– x)2 f(– x) = x4 – 4x2 (– 2, 0) (2, 0) (– 1, – 3) (1, – 3)

Unit 6 Lesson #1 Intercepts and Symmetry An odd function satisfies f (– x) = – f ( x ) (x, y) (-x, -y) The graph of an odd function is symmetric about the origin.

Unit 6 Lesson #1 Intercepts and Symmetry EXAMPLE 6 f(x) = x3 – 9x Show that this is an odd function. It is symmetrical to the origin An odd function satisfies f (– x) = – f ( x ) f(-x) = (-x)3 – 9(-x) – f(x) = – (x3 – 9x ) f(-x) = – x3 + 9x – f(x) = – x3 + 9x (– 2, 10) (2, – 10)

Unit 6 Lesson #1 Intercepts and Symmetry EXAMPLE 7 Determine if the function is even, odd, or neither. Find –f(x): Find f(-x): = 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 odd and symmetric about the origin

Unit 6 Lesson #1 Intercepts and Symmetry EXAMPLE 8 y = 4x2 – x Determine if the function is even, odd, or neither. Find –f(x): Find f(– x): ≠ 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