Unit 3 Review for Common Assessment
Feb 24, 2016
Unit 3 Review for Common
Assessment
Match the graph of a quadratic function with
it’s equation below:
f(x) = x2 f(x) = -(x+2)2+4 f(x) = (x+2)2-1
Describe the end behavior of the graph
of each given graph.
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x →∞, f (x) →∞
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x →−∞, f (x) →∞€
x →∞, f (x) →−∞
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x →−∞, f (x) →∞
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x →∞, f (x) →−∞
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x →−∞, f (x) →−∞
Use the Leading Coefficient Test to determine the end behavior of
the graph of the given polynomial function.
1.) f(x) = -x3 + 4x
2.) f(x) = x4 – 5x2 +4
3.) f(x) = x5 - x
5.) f(x) = -2x4 + 2x24.) f(x) = x3 – x2 - 2x
Rise Left, fall right
Rise left, rise right
Fall left, rise right
Fall left, rise right Fall left, fall right€
x →∞, f (x) →∞x →−∞, f (x) →∞
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x →∞, f (x) →−∞x →−∞, f (x) →∞
EVEN
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x →∞, f (x) →∞x →−∞, f (x) →−∞
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x →∞, f (x) →∞x →−∞, f (x) →−∞
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x →∞, f (x) →−∞x →−∞, f (x) →−∞
Determine without graphing, the critical
points of each function.
1.) f(x) = (x + 2)2 - 3 2.) f(x) = -x2 + 6x - 8
Min (-2,-3) Max (3,1)3.) f(x) = 3x3 - 9x + 5 4.) = x3 + 6x2 + 5x
Min (-.47, -1.13)Max (-3.53, 13.12)Pt. of Inflection (-2,6)
Min ( 1, -1)Max (-1, 11)Pt. of Inflection ( 0 , 5)
5.) f(x) = x4 - 10x2 + 9Min ( -√5, -16)Max (0, 9)Min ( √5 , -16)
f’(x) = 2x + 4 f’(x) = -2x + 6
f’(x) = 9x2 - 9 f’’(x) = 18x
Find the zeros of each polynomial
function.
1.) x2 – 40 = 0 2.) x3 + 4x2 + 4x = 0
3.) x2 + 11x – 102 = 04.) x2 + ¾x + ⅛ = 0
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x = ±2 10 x = 0, -2, -2
x = -17, 6 x = -½, -¼
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x 2 = 40
x = 40
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x(x 2 + 4x + 4) = 0x(x +2)(x +2) = 0
If you can’t figure it out then use Quadratic Formula
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(x +17)(x −6) = 0
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x + 12( ) x + 14( ) = 0
Find the zeros of the polynomial function by factoring.
1.) f(x) = x3 + 5x2 – 9x - 451.) f(x) = x3 + 4x2 – 25x - 100
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x 2(x + 4) −25(x + 4) = 0
(x 2 −25)(x + 4) = 0(x −5)(x +5)(x + 4) = 0
x = 5, -5, -4
Which of the following is a rational zero of
f(x) = –2x5 + 6x4 + 10x3 – 6x2 – 9x + 4 1, -3, -2, 4, -1 ????
Remember you could use synthetic division or just do p(x) and see if you get a remainder of ZERO
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p(4) = −2(4)5 +6(4)4 +10(4)3 −6(4)2 −9(4) + 4 = 0
So 4 is a factor, the others are not
OR
Use synthetic division to divide x4 + x3 – 11x2 – 5x +
30 by x - 2 . Then divide by x + 3 Use the result to find
all zeros of f(x).
x2 x C RSo you are left with: x2 - 5
Then all the zeroes are: -3, , 2
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± 5
List all possible rational zeros of
1.) 2.)
List all possible rational roots, use synthetic division to
find an actual root, then use this root to solve the
equation.
f(x) = 2x4 + x3 – 31x2 – 26x + 24
Hint 4 and -3/2 are roots
2x2 + 6x – 4USE QUADRATIC FORMULA!!!
Find the number of possible positive, negative, and imaginary
zeros of: 2,0 positive roots
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f (−x) = x 2 + 4x +5
0 negative roots
P N I P N I
P N I P N I
2
0
0
0
0
2
1 positive root
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f (−x) = x 4 −2x 3 + x 2 +2x −2
3,1 negative roots
1
1
3
10
2
3,1 positive roots
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f (−x) = 6x 4 + x 3 + 4x 2 + x −2
1 positive root 3
1
1
1
0
2
3,1 positive roots
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f (−x) = −5x 5 −6x 4 +24x 3 +20x 2 − 7x −2
2,0 positive roots
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Use the given root to find the solution set of the polynomial
equation.p(x) = x4 + x3 – 7x2 – x + 6GIVEN -3 IS A ROOT
Then we can find the rest by factoring:
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x 3 −2x 2 − x +2
x 2(x −2) −1(x −2)
(x 2 −1)(x −2)(x +1)(x −1)(x −2)
So the roots are:-3, -1, 1, and 2
Which equation represents the graph of
the function? f(x) = 2x2+2x-1f(x) = -x2-3x+4 f(x) = x2+10x-1
Approximate the real zeros of each
function.
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R(x) = 3x 4 + x 2 −1
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F(x) = x 3 − 4x +6
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H(x) = 2x 3 − 4x 2 − 3
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G(x) = x 2 + 3x +1
0.7, -0.7 -2.5
2.3 -0.4 and -2.6
Use the given root to find the solution set of the polynomial
equations
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x 4 + x 3 −8x 2 + 4x − 48
2i€
x 4 − 3x 3 −12x 2 +54x − 40
3-iSince 2i is a root, so is -2i
Turn the roots into factors, multiply them together, then use long division
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(x −2i)(x +2i) = x 2 + 4
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x 2 + 4 x 4 + x 3 −8x 2 + 4x − 48x 2 + x −12
)
Then factor to find the remaining roots
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x 2 + x −12 = (x − 3)(x + 4)So the roots are: 2i, -2i, 3, and -4
Since 3-i is a root, so is 3+i
Turn the roots into factors, multiply them together, then use long division
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(x − (3 − i))(x − (3+ i)) = x 2 −6x +10
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x 2 −6x +10 x 4 − 3x 3 −12x 2 +54x − 40x 2 + 3x − 4
)
Then factor to find the remaining roots
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x 2 + 3x − 4 = (x −1)(x + 4)So the roots are: 3-i, 3+I, 1, and -4
Find the vertical asymptotes, if any, of the graph of each
function.
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R(x) =x
x 2 − 4
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F(x) =x + 3x − 4
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H(x) =x 2
x 2 +1
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G(x) =x 2 −9
x 2 + 4x −21
x = -2, x = 2 x = 4
No vertical asymptote x = -7
Find the horizontal asymptote, if any, of the
graph of
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R(x) =x
x 2 − 4
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F(x) =x + 3x − 4
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H(x) =x 2
x 2 +1
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G(x) =3x 4 − x 2
x 3 − x 2 +1
y = 0 y = 1
y = 1 y = 3x + 3€
x 3 − x 2 +1 3x 4 +0x 3 − x 2 +0x +03x + 3
)
If a monomial is on bottom then you just break it up.Otherwise must do long division
Choose the correct graph for the rational
function
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R(x) =x 2 −1x
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F(x) =2x 2 −5x +2x 2 − 4
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H(x) =x 2
x 2 +1
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G(x) =2(x +2)2(x −5)(x +5)(x −2)2