1. 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

2. Unit 6 Lesson #1 Intercepts and Symmetry
EXAMPLE 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)

3. 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)

4. 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

5. 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

6. 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)

7. 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)

8. 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.

9. 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)

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

11. 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