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Algebra Review 3 - Fall 2008 - MAT 140, Exams of Algebra

Murray State University Algebra

A college algebra review for mat 140, fall 2008. It includes problems on determining the truth of algebraic statements, finding quadratic functions from graphs, maximizing revenue, enclosing rectangular plots, identifying polynomial functions, forming polynomials, and analyzing polynomial functions and rational functions. Students are expected to solve problems related to discriminants, vertex locations, unit pricing, fencing, and graphing.

Typology: Exams

2009/2010

Uploaded on 02/24/2010

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COLLEGE ALGEBRA - MAT 140

FALL 2008 - Review 3

Name :..........................................

I. State whether each statement is True or False as stated. Provide a clear reason for your answer.

•If the discriminant b2−4ac = 0, the graph off(x) = ax2+bx +c, a 6= 0, will touch the x−axis

at its vertex.

•The graph of f(x) = x2(x−3)(x+ 4) has exactly three x−intercepts.

•End behavior: the graph of the function f(x) = 3x4−6x2+ 2x+ 5 resembles y=x4for large

values of |x|.

•The graph of a function may intersect a vertical asymptote.

•The graph of a function will never intersect an oblique vertical.

II. Determine the quadratic function whose graph is given in Figure 1.

.....

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−1 1 2 3

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−4

−2

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•

vertex(1,−3)

(3,5)

Figure 1:

III. Suppose that the manufacture of a gas clothes dryer has found that, when the unit price is p

dollars, the revenue R(dollars) is R(p) = −4p2+ 4000p.

•What unit price for the dryer should be established to maximize revenue?

Algebra Review 3 - Fall 2008 - MAT 140, Exams of Algebra

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COLLEGE ALGEBRA - MAT 140

FALL 2008 - Review 3

Name :..........................................

I. State whether each statement is True or False as stated. Provide a clear reason for your answer.

• If the discriminant b

− 4 ac = 0, the graph off (x) = ax

+ bx + c, a 6 = 0, will touch the x−axis

at its vertex.

• The graph of f (x) = x

(x − 3)(x + 4) has exactly three x−intercepts.

• End behavior: the graph of the function f (x) = 3x

− 6 x

+ 2x + 5 resembles y = x

for large

values of |x|.

• The graph of a function may intersect a vertical asymptote.

• The graph of a function will never intersect an oblique vertical.

II. Determine the quadratic function whose graph is given in Figure 1.

x

y

vertex(1, −3)

Figure 1:

III. Suppose that the manufacture of a gas clothes dryer has found that, when the unit price is p

dollars, the revenue R (dollars) is R(p) = − 4 p

+ 4000p.

• What unit price for the dryer should be established to maximize revenue?

• What is the maximum revenue?

IV. A farmer with 4000 meters of fencing want to enclose a rectangular plot that borders on a river.

If the farmer does not fence the side along the river, what is the largest area that can be enclosed?

V. Determine which functions are polynomial functions. For those that are, state the degree.

(i) h(x) = 3 −

x (ii) F (x) =

x

x

(iii) G(x) = 2(x − 1)

(x

VI. Form a polynomial whose zeros and degree are given.

Zeros : − 1 , multiplicity 1; 3 , multiplicity 2; degree 3

VII. For the polynomial: f (x) = −x

(x

− 1)(x + 1)

a) Find the degree of the polynomial. Determine the end behavior; that is, find the power

function that the graph of f resembles for large values of |x|.

b) Find the x− and y−intercepts of the graph of f.

c) Determine whether the graph crosses or touches the x−axis at each x−intercept.

d) Use a graphing utility to determine the number of turning points on the graph of f. Approx-

imate the turning points, if any exist, round to two decimal places.

e) Use the information obtained in parts (a) to (d) to draw a complete graph of f by hand.

f) Find the domain of f. Use the graph to find the range of f.