1. Warm-Up
• Sketch the graphs of the following:
f(x)=x f(x)=x2 f(x)=x3 f(x)=x4 f(x)=x5
End Behavior:
Even functions either start up and end up or start down and end down
Odd functions either start down and end up or start up and end down.
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2. Match the equations with their graph.
f ( x) = x − 5
f ( x) = x 2 − 4 x
f ( x) = −3x 4 + 5 x 2
f ( x) = −2 x 5 + x 4 − 2 x3 + 5 x − 2
1
f ( x) = − x 2 + 3x − 5
2
f ( x) = x3 − x 2 + 3x − 6
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3. Section 2.2
A polynomial function is a function of the form
f ( x) = an x n + an −1 x n −1 +  + a1 x + a0 , an ≠ 0
where n is a nonnegative integer and each ai is a real number.
The polynomial function has a leading coefficient an and degree n.
Examples: f ( x) = − 2 x5 + 3 x 3 − 5 x + 1
f ( x) = x3 + 6 x 2 − x + 7
f ( x) = 14
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4. Solve the following 0 = x − 3x + 2 2
There are multiple ways to write the answers.
0 = x − 3x + 2
2
0 = ( x − 1)( x − 2) x=1 is a zero
x −1= 0 x − 2 = 0 x=1 is a solution
x =1 x=2 x-1 is a factor
(1,0) is an x-intercept
The correct ways depends on the question.
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5. A real number a is a zero of a function y = f (x)
if and only if f (a) = 0.
Real Zeros of Polynomial Functions
If y = f (x) is a polynomial function and a is a real number then
the following statements are equivalent.
1. x = a is a zero of f.
2. x = a is a solution of the polynomial equation f (x) = 0.
3. (x – a) is a factor of the polynomial f (x).
4. (a, 0) is an x-intercept of the graph of y = f (x).
A polynomial function of degree n has at most n zeros.
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6. Example: Find all the real zeros of
f (x) = x 4 – x3 – 2x2.
Factor completely:
f (x) = x 4 – x3 – 2x2
= x2(x2 – x – 2) y
= x2(x + 1)(x – 2)
2
The real zeros are x = -1,x=0 (–1, 0) (0, 0)
double root, and x = 2. x
–2
(2, 0)
When the roots are real the zeros
correspond to the x-intercepts. f (x) = x4 – x3 – 2x2
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7. Graphing Utility: Find the zeros of f(x) = 2x3 + x2 – 5x + 2.
10
– 10 10
Calc Menu:
– 10
The zeros of f(x) are x = – 2, x = 0.5, and x = 1.
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8. Solve for the zeros using a graphing calculator.
1. y = 3 x + 4 x − 15 x − 203 2
2. y = x − 13 x + 5 5 2
3. y = − x − 3 x + 8 x 3 2
4. y = x + x − 8 x − 12 3 2
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9. Write the polynomial with the following roots.
1. x = 3, −2
2. x = ± 3, 0
3. x = 2 ± 5, −4
4. x = 3 double root , −2, 0
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