Download Pre-Test 4 (Chapters 5 – 6)

Pre-Test 3
Chapters 4, 5
Section 4.1
1.
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2.
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Section 4.2
3. Determine whether the given function is linear or nonlinear. If it is linear, determine the
equation of the line.
(a)
x
1
5
10
12
f ( x)
8
16
26
32
Nonlinear
(b)
x
1
2
4
5
f ( x)
3
5
9
11
5  3 9  5 11  9


 2 linear
2 1 4  2 5  4
(1,3) : 3  2(1)  b; b  1, y  2 x  1
m
4. For the given data points below use the calculator to find the best fit line.
x
6
8
10
12
f ( x)
1
4
6
10
y  1.45 x  7.8
Section 4.3
5. For the quadratic equation f ( x)  2 x2  12 x  22 , find the following:
a. Does it open up or down?
b. Find the vertex.
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c.
d.
e.
f.
Graph the function.
Find the y-intercept and x-intercept(s).
Find the domain and range.
Determine where the function is increasing and decreasing.
(a) Up
(b)
b (12) 12

  3,
2a
2(2)
4
f (3)  4;
(3, 4)
(c)
y
9
8
7
6
5
4
(3,4)
3
2
1
x
1
2
3
4
5
(d) f(0) = 22; y-intercept: (0,22);
No x-intercepts
(e) Domain: All Reals,
Range: y ≥ 4
(f) Increasing: (3, ∞);
Decreasing: (-∞, 3)
6. Determine algebraically, without graphing, the minimum value of the quadratic
function
f ( x)  2 x 2  3x  2 .
b (3) 3

  .75,
2a 2(2) 4
f (.75)  .875;
(.75, .875)
Section 4.4
7. Maximizing Revenue The price p ( in dollars) and the quantity x sold of a certain
product obey the demand equation
p
1
x  15
10
(a) Express the revenue R as a function of x.
 1
 1 2
R( x)  xp  x  x  15  
x  15 x
 10
 10
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(b) What is the revenue if 20 units are sold?
R(20) 
1
(20) 2  15(20)  260
10
(c) What quantity x maximizes revenue? What is the maximum revenue?
b
15

 75
2a
 1
2  
 10 
1
2
R(75)   75   15(75)  $562.50
10
(d) What price should the company charge to maximize revenue?
p
1
(75)  15  $7.50
10
Section 5.1
1. Determine which functions are polynomial functions. For those that are, state the
degree.
(a) f(x) =
5x2
+
4x4
Yes, degree 4
(b) F(x) =
x2  5
x3
No
2. Form a polynomial function whose real zeros and degree are given.
a. Zeros: -2, 3,4; degree 3
f(x) = (x + 2)(x – 3)(x – 4) =
( x 2  x  6)( x  4)  x3  5x 2  2 x  24
b. Zeros: -1, multiplicity 1; 3, multiplicity 2; degree 3
f ( x)  ( x  1)( x  3)2  ( x  1)( x 2  6 x  9)  x3  5 x 2  3x  9
3. For the polynomial function f(x) = 4(x + 4)(x + 3)3:
(a) List each real zero and its multiplicity
(b) Determine whether the graph crosses or touches the x-axis at each x-intercept.
(c) Determine the maximum number of turning points for the graph.
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(d) Determine the end behavior; that is, find the power function that the graph
resembles for large values of |x|.
4. Analyze the polynomial function f(x) = (x – 1)(x + 3)2 by finding the following:
(a) Determine the end behavior of the graph of the function.
(b) Find the x- and y-intercepts of the graph of the function.
(c) Determine the zeros of the function and their multiplicity.
(d) Use a graphing calculator to graph the function.
(e) Find the turning points of the graph.
(f) Find the domain and range of the function.
(g) Use the graph to determine where the function is increasing and where it is
decreasing.
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Section 5.2
5. Find the remainder when
f ( x)  3x 4  6 x3  5x  10 is divided by x - 2. Then use
the Factor Theorem to determine whether x - 2 is a factor of f(x).
6. Determine the maximum number of real zeros that
f ( x)  x5  x 4  2 x 2  3 may
have. Then list the potential rational zeros of the function. Do not attempt to find
the zeros.
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7. Find the bounds to the zeros of each polynomial function.
a.
b.
8. Find the real zeros of x  x  2 x  4 x  8 
synthetic division to factor the function.
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2
8
0.
Use the real zeros and
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Section 5.3
9. Information is given about a polynomial f(x) whose coefficients are real numbers.
Find the remaining zeros of f.
a. Degree 3; zeros: 4, 3 + i 3 - i
b. Degree; 6
zeros: 2, 2 + i, -3 – i, 0
2 – i -3 + i
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10. Form a polynomial f(x) with real coefficients having the following degree and zeros.
a. Degree 4;
zeros: 2 + i, and 3, multiplicity 2
f ( x)  ( x  (2  i))( x  (2  i))( x  3)( x  3)
f ( x)  ( x  (2  i))( x  (2  i))( x 2  6 x  9)
f ( x)  ( x  2  i ))( x  2  i ))( x 2  6 x  9)
f ( x)  ( x 2  2 x  xi  2 x  4  2i  ix  2i  i 2 )( x 2  6 x  9)
f ( x)  ( x 2  4 x  5)( x 2  6 x  9)
f ( x)  x 4  6 x 3  9 x 2  4 x 3  24 x 2  36 x  5 x 2  30 x  45
f ( x)  x 4  10 x 3  38 x 2  66 x  45
Section 5.4
11. Find the domain of each rational function.
(a) R( x) 
5x2
x3
(b) Q( x) 
 x(1  x)
3x 2  5 x  2
12. Find the vertical, horizontal, and oblique asymptotes, if any, for each rational
function.
(a) R( x) 
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3x  5
x6
(b) F ( x) 
x2  6 x  5
2x2  7 x  5
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Section 5.5
13. Analyze the rational functions below by finding the following:
R( x) 
a.
b.
c.
d.
e.
f.
g.
h.
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( x  1)( x 2  x  1)
x
G ( x) 
x2  2 x
( x  1)( x  2)
Factor the numerator and denominator. Find the domain.
Write the function in lowest terms.
Find the x- and y-intercepts of the graph.
Test for symmetry. Is it symmetry with respect to the y-axis, origin, or neither.
Local the vertical asymptotes.
Local the horizontal or oblique asymptotes, if any.
Graph the function using a graphing utility.
Draw a complete graph of the function by hand using the information from
steps a – g.
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14. Drug Concentration The concentration C of a certain drug in a patient’s
50t
bloodstream t minutes after injections is given by C (t )  2
.
t  25
a. Find the horizontal asymptote of C(t). What happens to the concentration of
the drug as t increases?
b. Using your graphing utility, graph C = C(t).
c. Determine the time at which the concentration is highest.
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