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1 the x-intercepts of y = f (x), set y = 0 and solve f INTERCEPTS AND ZEROS To find the y-intercepts of y = f (x), set x = 0; the y- intercept is f (0). tercepts 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

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

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Page 1: 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

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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) = x2 – x – 2 x2 – 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

Page 2: 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

2

EXAMPLE 1 f(x) = 3x2 + 7x – 6

x-intercepts (Set y = 0)

3x2 + 7x – 6 = 0

3x2 + 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

Page 3: 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

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EXAMPLE 2 f(x) = 2 sin x, -p < x < p

x-intercepts (Set y = 0) y-intercept (Set x = 0)

  

 

 

Also

 

   

 

q

(0, 1)

(0, – 1)

y

1

 

(1, 0)x(– 1, 0)

 

𝐬𝐢𝐧(−𝝅 )=𝟎

Page 4: 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

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EXAMPLE 3 f(x) = 2cos x – 1, 0 < x < 2p

x-intercepts (Set y = 0) y-intercept (Set x = 0)

  

1

2

2

 

𝒙=𝟑𝟎𝟎°=𝟓𝝅𝟑

Page 5: 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

5

SymmetryAn even function satisfies f (– x) = f ( x ).

The graph of an even function is symmetric about the y-axis

( x , y)(– x , y)

Page 6: 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

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EXAMPLE 4 f(x) = 9 – x2

f(– x) = 9 – (– x)2

f(– x) = 9 – x2

Show that this is an even function. It is symmetrical to the y-axis

(2, 5)(– 2, 5)

(– 3, 0) (3, 0)

Page 7: 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

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EXAMPLE 5 f(x) = x4 – 4x2

f(– x) = (–x)4 – 4(– x)2

f(– x) = x4 – 4x2

Show that this is an even function. It is symmetrical to the y-axis

(– 2, 0) (2, 0)

(1, – 3)(– 1, – 3)

Page 8: 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

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The graph of an odd function is symmetric about the origin.

An odd function satisfies f (– x) = – f ( x )

(x, y)

(-x, -y)

Page 9: 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

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EXAMPLE 6 f(x) = x3 – 9x

Show that this is an odd function. It is symmetrical to the origin

– f(x) = – (x3 – 9x )

– f(x) = – x3 + 9x

An odd function satisfies f (– x) = – f ( x )

(– 2, 10)

(2, – 10)

f(-x) = (-x)3 – 9(-x)

f(-x) = – x3 + 9x

Page 10: 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

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EXAMPLE 7

Determine if the function is even, odd, or neither.

51( )

2f x x

Find f(-x):

51( ) ( )

2f x x

Since f(x) ≠ f(– x), the functionis not even and not symmetricabout the y-axis.

Find –f(x):

51( )

2f x x

51( )

2f x x

Since f(– x) = – f(x) the functionis odd and symmetric aboutthe origin

51( )

2f x x

=

Page 11: 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

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EXAMPLE 8 y = 4x2 – x

Determine if the function is even, odd, or neither.

Since f(x) ≠ f(–x), the functionis not even and not symmetricabout the y-axis.

Since f(–x) ≠ – f(x) the functionis not odd and not symmetric about the origin

Find f(– x):

2( ) 4( ) ( )f x x x

2( ) 4f x x x

Find –f(x):

2( ) 4f x x x

2( ) 4f x x x ≠