COLLEGE ALGEBRA I (MATH 006)
SPRING 2015
——PRACTICE FOR EXAM IV——
1. A quadratic function is given.
(a) Express the function in vertex form
(f (x) = a(x − h)2 + k).
(b) Find its vertex and its x− and y−intercepts.
(a) f (x) = x2 − 6x
(b) f (x) = 2x2 + 4x + 3
2. Find the maximum or minimum value of the function.
(a) f (x) = x2 + 2x − 1
(b) f (x) = −x2 − 3x + 3
3. Solve the quadratic inequality.
x2 < 10 − 3x
1
4. Find f + g, f − g, f g, and f /g and their domains.
(a) f (x) = x − 3, g(x) = x2
(b) f (x) = x2 , g(x) =
4
x+4
5. Find the√domain
√ of the function.
f (x) = x + 1 − x
2
6. Find the functions f ◦ g, g ◦ f, and f ◦ f .
(a) f (x) = x2 , g(x) = x + 1
(b) f (x) =
x
,
x+1
g(x) = 2x − 1
7. The graph of a polynomial function is given.
(a) Find all local maximum and minimum values of the function and the value of x at which each occurs.
(b) Find the intervals on which the function is increasing and
on which it is decreasing.
(c) What is the sign of the leading coefficient of the function(+
or −)?
(d) Is the degree of the function even or odd?
y
3
2
1
−2
−1
−1
1
2
x
−2
−3
3
8. Factor the polynomial and use the factored form to find the
zeros. Then sketch the graph.
P (x) = x2 − 4
9. Solve for x
(3x + 12)(x2 − 11x + 28) = 0
10. Find the quotient and remainder.
2
3
(a) x −6x−8
(b) xx2 +6x+3
x−4
−2x+2
4
11. For the polynomial P (x) = 2x3 + 5x2 − x − 6
(a) Evaluate P (−2)
(b) Use synthetic(or long) division to show that x+2 is a factor
of P (x).
(c) Use the result from (b) to write P (x) in fully factored form.
12. Solve the polynomial inequality.
x3 − x2 − 6x ≥ 0
5
13. Express the statement as an equation, and find the constant of
proportionality.
(a) y is directly proportional to x. If x = 6, then y = 42.
(b) R is inversely proportional to s. If s = 4, then R = 3.
(c) F varies jointly as the square of c and d. If c = 30 and
d = 20, then F = 150.
(d) M varies directly as x and inversely as y. If x = 2 and
y = 6, then M = 5.
14. Solve for x.
(a) 53x = 54x−2
(b) (1 − x)5 = (2x − 1)5
(c) 42x+7 = 8x+2
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15. If $10, 000 is invested at an interest rate of 3% per year, compounded semiannually, find the value of the investment after
the given number of years.
(a) 5 years
(b) 10 years
16. If $2000 is invested at an interest rate of 3.5% per year, compounded continuously, find the value of the investment after
the given number of years.
(a) 2 years
(b) 4 years
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17. Uranium-240 has a half-life of 14 hours. The amount A(t) of
a sample of uranium-240 remaining (in grams) after t hours is
given by
a(t) = 3900
1
2
t
14
(a) Find the initial amount in the sample.
(b) How much remains after 30 hours?
18. The population P (t) of bacteria cells in a petri dish after t
hours is given by
P (t) = 2000(0.86)t
(a) Find the initial population size.
(b) Does the function represent growth or decay?
(c) By what percent does the population size change each
hour?
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