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1.5B Polynomial functions and complex zeros

Worksheet
What do you remember?
1

Define an even function. How is it graphically represented?

2

Define an odd function. How is it graphically represented?

3

What does the property f (−x) = f (x) signify about a function?

4

Consider the function p(x) = a_n x^n, where n ≥ 1 and an ≠ 0. If n is even, is the function even or odd?

Let's practice
5

Determine if the following functions are even, odd or neither based on their symmetry:

a

f \left( x \right) = x^3 - x

b

f \left( x \right) = x^4 - 3x^2 + 2

c

f \left( x \right) = x^5 - 4x^3 + 2x

d

f \left( x \right) = x^2 + 2x + 1

e
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
f
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
6

For each of the given functions, state whether the graph is symmetric about the y-axis, the x-axis, the origin, or is not symmetric:

a

f \left( x \right) = x^2 - 4x + 4

b

f \left( x \right) = x^3 + 3x^2 - x - 3

c

f \left( x \right) = x^3 - 6x

d

f \left( x \right) = 2x^4 - 8x^2 + 2

7

Determine whether the following polynomial functions are even, odd, or neither:

a
f(x) = x^4 - 2x^2 + 1
b
f(x) = 3x^5 - x^3 + 7x
c
f(x) = x^6 + 4x^4 - 9x^2
d
f(x) = 2x^7 - 5x^5 + 3x^3
e
f(x) = x^8 - 6x^6 + 9x^4 - 4x^2
f
f(x) = 5x^7 - 3x^5 + x^3 - 9x
g
f(x) = x^3 - x^2 + x - 1
h
f(x) = 4x^8 - 3x^6 + 2x^4 - x^2
8

Given the graph of a polynomial function below, determine if the function is even, odd, or neither:

-4
-3
-2
-1
1
2
3
4
x
-18
-16
-14
-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
14
16
18
y
9

Determine whether the following graphs are even, odd, or neither. Justify your answer.

a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
c
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
10

A polynomial function has the following properties:

  • It is a cubic function.

  • The function is symmetric about the origin.

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

Let's extend our thinking
11

Given the polynomial function f(x) = 4x^5 - 2x^3 + x, determine whether it is even, odd, or neither. Justify your answer using the properties of even and odd functions.

12

The function f(x) = x^4 - 2x^2 + 1 is said to be even. Provide a demonstration to justify this statement.

13

The function f(x) = x^3 - 3x is said to be odd. Provide a demonstration to justify this statement.

14

Can a function be both even and odd at the same time? Justify your answer.

15

Consider the function f(x) = 3x^4 - 5x^2 + 2. Is this function even, odd or neither? Justify your answer.

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Outcomes

1.5.B

Determine if a polynomial function is even or odd.

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