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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
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
3
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
4
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
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)
6
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)
7
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)
8
The graph of an odd function is symmetric about the origin.
An odd function satisfies f (– x) = – f ( x )
(x, y)
(-x, -y)
9
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
10
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
=
11
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 ≠