Algebra / Graphing Review
AGEC 317
Material from MATH 141 and 142
The ability to manipulate algebraic expressions and provide graphical analysis is
critical in economic studies. The following review problems provide examples of
equation manipulation and graphing. To get the full benefit of the review, do not use a
calculator / spreadsheet.
1.
(a)
2.
9 x 2  21x
2x  9
(b)
6 2
x
42
(b)
1x 1
7 x a b
(c )
9x  7
2x  9
(d )
9 x 2  21
3 x
1 1 a b
x
7
1
( d ) x 7  a b
7
(e) unable to simplify
(c ) 2  2 x  3 y
(d )
3( y  2 x)
5(2 x  3 y )
If f ( x)  2  3x and g ( x)  3  4 x what does f ( x) g ( x)  ?
(b) 5  7 x
(c) 6  x  12 x 2
(d )
2  3x
3  4x
A water tank is initially 1/4 full. After adding 33 gallons of water, the water tank
is 4/5 full. What is the capacity of the water tank?
(a) 25
6.
(c )
(b) 8  4 y  3x
(a) 6  17 x  12 x 2
5.
(c ) 6 x  8
Suppose that y  2 x  3 and x  3 y  5, what does x + y = ?
(a) 2  2 x  3 y
4.
3x  7
1
3x
6x3
Which of the following is equivalent to
?
42 x a 4b
(a)
3.
3x  7
?
2 3

3 x
Which of the following is equivalent to
(b) 34
Simplify 125
(a) 25
(c) 50
(d ) 82
(e) 60
2
3
(b)
1
25
(c) 5,208
(d )
1
5
(e) who cares
7.
Suppose x  3 y  5 and 2 x  4 y  3 y, solve for x and y.
(a) x  1, y  2
8.
(b) x  1, y  2
(c) e 2 x  e 2 x
(b) e
(d ) e 2 x  e 2 x
(b) 6
(c ) 
5
6
(d )
6
5
( e)
1
6
What is the intercept given by the line in question 9?
(a) 5
11.
y2
What is the slope of the line in (x, y space) given by x  5  6 y ?
(a) 5
10.
(d ) x  1,
Expand and simplify (e x  e  x )(e x  e  x ) where e is the natural exponential
function.
(a) 2  e 2 x  e 2 x
9.
(c) x  2, y  1
(b) 6
(c ) 
5
6
Let f ( x)  4 x 2  3, and g ( x) 
(a ) f (3)
(b) g(2)
(d )
6
5
( e)
1
6
4
x  x  2 , find
5
(c) f (1)g(25)
(d) x if f ( x )  61
12.
If total costs to produce 5 units of good A are $1250 and fixed costs are $250,
what is the linear cost function associated with good A?
13.
Graph the following functions y  3, x  2, and x  4 
14.
Graph the following function f ( x)  3  10 x 2 .
15.
Suppliers will market 500 books when the per unit price is $25 and will market
$1000 books when the unit price is $30. Consumers will demand 10,000 books if
they are free. For each dollar rise in the price of a book, consumers will demand
200 fewer books.
(a) Find the demand and supply equations.
(b) Calculate the equilibrium point for the book market.
(c) Graph the equations and note the equilibrium point
4
y in x, y space.
5
16.
Graph the following functions and shade the feasible region (region that satisfies
and the functions):
x  1, y  1, y  6  1x, y  3 .
17.
What are the corner points in the graph from question 16? What is the
significance of the corner points?
18.
Graph the function y = ln x.
Answers and Hints
1.
a - to add fractions you need a common denominator, recall how a fraction in the
denominator can be moved to the numerator
2.
d - to move a variable from the denominator to the numerator multiple the
exponents by -1, to combine variables that are multiplied, add the exponents
3.
c - solve for y in the first equation and for x in the second equation, then add the
equations. To move a variable from one side of the equation to the other, subtract
or add it to both sides of the equation
4.
a - multiply each component in the first function, f(x), by the components in the
second function, g(x)
5.
e - set up an algebraic equation such that the initial volume in the tank plus
gallons added equals the ending full. Use the capacity of the tank as the variable
to be solved.
6.
b - negative exponent moves the variable to the denominator, a fraction such as
1/3 is taking the cube root.
7.
b - solve one equation for x, substitute for x in the other equation and solve for y,
put your numerical value for y in either equation to get the x value.
8.
c
9.
e - be sure to put the equation in the form y  a  bx where a is the intercept and
b the slope.
10.
c
11.
(a) f(x) = 33
(b) g(x) = 27/20 = 1.35
(c) f(x)g(x) = 20-1/(25)2=19.9984
(d) x = 4
Hint: f(3) is the notation for substituting in 3 for x in the equation f(x).
12.
TC = $250 + $200 A, where A is the number of units of good A produced. Hint,
fixed costs is the intercept and solve the TC equation for 5 units of A for the
coefficient associated with producing each unit of A.
13.
x = 4 - 4/5y is a downward sloping line with a y-intercept at y = 5 and an xintercept at x = 4 (be sure to put the equation in x, y space), y = 3 is a horizontal
line at y = 3, x = 2 is a vertical line at x = 2.
y
x= 2
y=3
x
y = 5 - 5/4 x
14.
Concave function with a maximum value at x = 0 and y = 3.
f(x)
x
15.
(a) qx = -2000 + 100 p, (hint: set up two equations with parameters for the slope
and intercept. Solve the two equations for the two unknown parameters), qd =
10,000 - 200 p
(b) equilibrium price and quantity of books are p = 40 and q = 2000.
(c) to graph put quantity on x-axis and price on the y-axis. Equilibrium point is
where supply and demand cross (equal).
Price
Supply
10,000
(2000, 40)
20
Quantity
50
Demand
-2,000
16.
feasible area shaded, must satisfy all constraints
6
3
1
1
6
17.
Corner points are points where the functions determining the feasible region
cross. They are (1,1), (1,3), (3,3), (5,1) and are found by solving each set of
equations that cross simultaneously. Corner points will give the maximum or
minimum point to an appropriate defined objective function that meets all
constraints.
18.
Function is asymptotic to both the x- and y-axis and is undefined for x ≤ 0.