Revised 12/20/04

MATH 115: Final Exam Review Can you solve an inequality algebraically? (1.1) (1) Solve the inequality 13 2 5 9x− ≤ − − < . Express the solution in

interval notation, and graph the solution set. Can you find the distance between two points and the midpoint of a line segment? (1.2) (2) Consider the points ( )3,5A − and ( )6, 2B − . (a) Find the distance between A and B.

(b) Find the coordinates of the midpoint of AB . Can you find the x- and y-intercepts of the graph of an equation algebraically? (1.3) (3) Find all x- and y-intercepts of the graph of each equation.

(a) 25 3 12x y+ = (b) 2 9x

yx−

=

Can you determine whether the graph of an equation will be symmetric with respect to the x-axis, y-axis, or origin algebraically? (1.4)

(4) Determine whether the graph of 2 9x

yx−

= is symmetric with

respect to the x-axis, y-axis, or origin. Can you graph a circle by hand given the equation of the circle? (1.4) (5) Find the center ( ),h k and radius r of the circle

2 2 6 2 0x y x y+ + − = . Graph the circle by hand. Can you write the equation of a circle given information about the circle? (1.4) (6) Find the standard form of the equation of the circle with

endpoints of a diameter at ( )1, 4− − and ( )5,7 . Can you use a graphing utility to solve an equation? (1.5) (7) Use a graphing utility to approximate the real solutions, if any,

of the equation rounded to two decimal places.

2 5x xπ= +

Revised 12/20/04

Can you algebraically find the domain of a function, identify the graph of a function, and obtain information from or about the graph of a function? (1.7)

(8) Consider ( ) 4

3

xf x

x=

+.

(a) Is the point ( )2,8− on the graph of f? (b) If 0x = , what is ( )f x ? What point is on the graph of f? (c) If ( ) 8f x = , what is x? What point is on the graph of f? (d) What is the domain of f? Can you determine characteristics of a function, including domain and range, average rate of change, intervals of increase and decrease, local maxima and minima, and whether the function is odd or even? (1.8) (9) Consider ( ) 2 4f x x x= − .

(a) Find the range of f. (b) Find the average rate of change of f from 2x = to 5x = . (c) Determine algebraically whether f is odd, even, or neither.

Can you find the sum, difference, product, quotient, and composition of two functions, and can you state the domains of each? (1.6 and 6.1)

(10) Find the composite function ( ) ( )f g x given ( ) 21

f xx

=+

and

( ) 1g x x= − . State the domain of ( ) ( )f g x . Can you graph a line given the equation of the line or write the equation of a line given information about the line? (2.2) (11) Write an equation of the line passing through ( )2, 9− and

perpendicular to the line 2 5x y+ = . Express your answer in general form and in slope-intercept form.

Revised 12/20/04

Can you use a graphing utility to find the line of best fit to a data set, and can you interpret the slope of the line of best fit? (2.4) (12) The table shows Test 3 scores vs. final course averages for a

particular class. Use a graphing utility to find the line of best fit to the data. Interpret the slope of the line of best fit.

Test 3 Score Final Course Average

70 79 78 89 67 76 65 79 57 68 86 97 77 85 84 92 56 79

Can you find the real solutions of a quadratic equation algebraically? (3.1) (13) Find the real solutions, if any, of 22 14 5x x− = by completing

the square. Can you solve an equation that is quadratic in form algebraically? (3.1)

(14) Find the real solutions, if any, of 2 5

6 02 2

x xx x

⎛ ⎞ + + =⎜ ⎟+ +⎝ ⎠.

Can you graph a quadratic function by hand and algebraically determine properties of the graph? (3.2) (15) Graph ( ) 2 3 7f x x x= − − by hand. Identify the vertex, the axis of

symmetry, and any intercepts. Can you perform operations on complex numbers and find the complex solutions of a quadratic equation? (3.5)

(16) Write 5 22

ii

+−

in the standard form a bi+ .

(17) Solve the equation 2 3 3 0x x+ + = in the complex number system. Can you solve a radical equation algebraically? (4.1)

(18) Find the real solutions, if any, of 5 3 2 3x x+ = − + . Can you solve an equation or inequality involving absolute value algebraically? (4.1) (19) Solve 5 2 7x− = .

Revised 12/20/04

(20) Solve 5 2 12x− ≥ . Express the solution in interval notation. Can you graph, evaluate, and determine characteristics of a piecewise-defined function? (4.2)

(21) Consider the function ( )if 1

1 if 1 3

x xf x

x x

− ≤ −⎧= ⎨− − − < ≤⎩

.

(a) State the domain of f. (b) Identify any intercepts of f. (c) Sketch a graph of f by hand. (d) Find the range of f using your graph. Can you graph a function using transformations? (4.3) (22) Given here is a complete graph of a function f. Use this graph to

sketch the graph of ( ) ( )2 1 1F x f x= + − . Can you graph transformations of a power function by hand? (5.1)

(23) Graph ( ) ( )53 1f x x= − + using transformations.

Revised 12/20/04

Can you analyze a polynomial function and its graph? (5.2)

(24) Consider the polynomial function ( ) ( ) ( )23 1 2f x x x= − + − . (a) Algebraically find the x- and y-intercepts of f.

(b) Determine for each x- intercept whether the multiplicity is even or odd, and thus determine whether the graph of f will touch the x-axis or cross the x-axis there.

(c) State the power function that the graph of f will resemble

for large values of x . (d) Confirm (a) – (c) using a graphing utility. (e) Determine the number of turning points on the graph of f. (f) Determine any local maxima and local minima of f. Can you solve polynomial and rational inequalities? (5.5)

(25) Solve the rational inequality 1 21 2 5

x xx x+

≤− +

algebraically.

Can you find the real zeros of a polynomial function? (5.6) (26) Use the Factor Theorem to determine whether 3x − is a factor

3 2 8 12x x x+ − − . If it is, write f in factored form. Can you find the complex zeros of a polynomial? (5.7) (27) Find the complex zeros of 4 3 23 11 53 60x x x x− − + − . Can you graph a parabola given the equation of the parabola or write the equation of a parabola given information about the parabola? (7.2)

(28) A parabola has vertex ( )2,3 and directrix 12

x = . Write the

equation of the parabola in standard form and graph the parabola by hand.

Can you graph an ellipse given the equation of the ellipse or write the equation of an ellipse given information about the ellipse? (7.3)

(29) An ellipse has equation ( ) ( )2 23 2

14 16

x y+ −+ = . Identify the

coordinates of the center, the vertices, and the foci of this ellipse, and graph the ellipse by hand.

Revised 12/20/04

Can you graph a hyperbola given the equation of the hyperbola or write the equation of a hyperbola given information about the hyperbola? (7.4) (30) A hyperbola has center ( )2, 1− , a vertex at ( )0, 1− , and a focus at

( )5, 1− . Write the equation of the hyperbola in standard form. Can you solve systems of linear equations, including systems that are inconsistent and systems that consist of dependent equations? Can you use a system of linear equations to model a problem mathematically? (8.1 and 8.2) (31) Solve the given system using substitution or elimination.

2 19

3 24

73

x y

x y

+ =

− =

(32) Solve the given system by elimination.

1

2 3 4

3 2 7 0

x y z

x y z

x y z

− − =− + − = −

− − =

(33) Solve the given system by performing row operations by hand on an

augmented matrix.

2 4 2 10

3 4 2 5

5 6 3 3

x y z

x y z

x y z

+ − = −− + − =

+ + =

(34) Write a system of equations to model this problem, and then use

the matrix features of your graphing utility to solve the problem.

Bob received a lottery check for $10,000. He invested the money into three different accounts. Part of his money was placed in a savings account paying 7% interest. A second portion, which was twice the first amount, was deposited in a CD paying 9% interest. The remainder of the money was put into a money market fund yielding 10% interest. If Bob’s total interest over a one-year period was $925, how much was deposited in each of the three accounts?

Revised 12/20/04

Can you perform operations on matrices? (8.4) (35) Without using a graphing utility, find AB, where A and B are as

given.

1 0 2

1 2 1A

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

4 0

1 2

2 3

B⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥− −⎣ ⎦

Can you set up and solve a system of nonlinear equations? (8.6) (36) A rectangle has area 120 in2 and perimeter 46 in. Find the length

of a diagonal of the rectangle. Can you find the nth term of a sequence and express a sum using summation (sigma) notation? (9.1) (37) Write down the apparent nth term of the sequence

1, 3, 5, 7, 9, ...− − . (38) Express the sum ( ) ( ) ( ) ( ) ( )2 2 1 2 4 2 9 2 16 ... 2 225+ + + + + + + + + + +

in summation notation. Can you find a formula for the nth term of an arithmetic sequence, and can you solve an application problem involving the sum of the terms of an arithmetic sequence? (9.2) (39) The 3rd term of an arithmetic sequence is 9− , and the 8th term of

the same sequence is 11. Find a formula for the nth term of the sequence.

(40) A trapezoidal seating area contains 50 seats in the first row.

Each successive row contains 2 fewer seats than the previous row. The last row contains 10 seats. How many seats are there in this seating area?

Can you find a formula for the nth term of a geometric sequence, and can you find the sum of a geometric series? (9.3) (41) Find an expression for the 42nd term of the sequence

1618, 12, 8, , ...

3 .

(42) Find the sum if it exists: 16

18 12 8 ...3

+ + + + .

Can you expand a binomial using the Binomial Theorem? (9.5)

(43) Expand ( )43 5x − using the Binomial Theorem.

Revised 12/20/04

Answers (with occasional explanations): (1) ( 7,4]x ∈ −

(2) (a) 130 (b) 3 3,

2 2⎛ ⎞⎜ ⎟⎝ ⎠

(3) (a) x-intercept 125; y-intercepts 2±

(b) x-intercepts 3± ; no y-intercept (4) Only replacing both x by x− and y by y− yields an equivalent

equation, so the graph is only symmetric with respect to the origin.

(5) Center = ( )3,1−

radius = 10 3.16≈

(6) ( )2

2 3 1572

2 4x y⎛ ⎞− + − =⎜ ⎟

⎝ ⎠

(7) { }1.52, 3.29x = −

(8) (a) no (b) If 0x = , ( ) 0f x = .

The point ( )0,0 is on the graph of f. (c) If ( ) 8f x = , 6x = . The

point ( )6,8 is on the graph of f. (d) ( )3,− ∞

Revised 12/20/04

(9) (a) [ 4, )− ∞

(b) ( ) ( ) ( )5 2 5 4

35 2 3

f f− − −= =

−

(c) ( ) ( ) ( )2 24 4f x x x x x− = − − − = + , which is neither ( )f x nor

( )f x− . So f is neither odd nor even.

(10) ( ) ( ) 2

1 1f g x

x=

− +. The domain is ( ,1]−∞ .

(11) 2 20x y− = (general form); 1

102

y x= − (slope-intercept form)

(12) 0.74 29.88y x≈ + . For every one-point increase in the Test 3

score there is, on average, approximately a 0.74-point increase in the final course average.

(13) 5 137 5 137

,4 4 4 4

x⎧ ⎫⎪ ⎪= − +⎨ ⎬⎪ ⎪⎩ ⎭

(14) { }3 4,

2 3x = − −

(15) The vertex is 3 37,

2 4⎛ ⎞−⎜ ⎟⎝ ⎠

.

The axis of symmetry is

32

x = . The x-intercepts are

3 372 2

x⎧ ⎫⎪ ⎪= ±⎨ ⎬⎪ ⎪⎩ ⎭

or

{ }1.54, 4.54x ≈ − .

The y-intercept is 7− .

(16) Multiplying by 22

ii

++

yields 8 95 5

i+ .

(17) 3 3 3 3

,2 2 2 2

x i i⎧ ⎫⎪ ⎪= − − − +⎨ ⎬⎪ ⎪⎩ ⎭

(18) 1x = − (19) { }1,6x = −

Revised 12/20/04

(20) 7 17

( , ] [ , )2 2

x ∈ −∞ − ∪ ∞

(21) (a) ( ,3]−∞ (b) no x-intercept; y-intercept is 1− (c)

(d) [ )4,0 [1, )− ∪ ∞

(22)

Revised 12/20/04

(23) Start with the graph of 5y x= . Shift the graph to the left 1 unit, reflect it over the x-axis, and then shift it up 3 units. (24) (a) x-intercepts: 1− and 2; y-intercept: 6

(b) 1− is of even multiplicity, so the graph will touch the x-axis there.

2 is of odd multiplicity, so the graph will cross the x-axis there.

(c) ( ) 33f x x= − (d) (e) 2 turning points (f) local minimum 0 at 1x = − ; local maximum 12 at 1x =

(25) 5 5

( , ) [ ,1)2 9

x ∈ −∞ − ∪ −

(26) ( )3 0f = , so 3x − is a factor of f. The factored form of f is

( ) ( )23 2x x− + . (27) { }4,3,2 ,2x i i= − − +

-4 -2 2 4

-10

10

20

30

Revised 12/20/04

(28) ( ) ( )23 6 2y x− = −

(29) Center ( )3,2− . Vertices ( )3, 2− − and ( )3,6− . Foci ( )3,2 2 3− ± .

(30) ( ) ( )2 22 1

14 5

x y− +− =

(31) 75 11

,8 2

⎛ ⎞⎜ ⎟⎝ ⎠

(32) There are infinitely many solutions, each of the form

( )2 5 , 3 4 , ,z z z z− + − + ∈ . (33) ( )3,1,4−

Revised 12/20/04

(34) Let s represent the amount invested in savings. Let c represent the amount invested in the CD. Let m represent the amount invested in the money market fund. The system we initially write is

10000

2

0.07 0.09 0.10 925

s c m

c s

s c m

+ + ==

+ + =.

Rearranging the second equation, we input the following augmented

matrix into the graphing utility:

1 1 1 10000

2 1 0 0

0.07 0.09 0.10 925

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

.

Using the feature that produces the reduced row-echelon form of a

matrix, we find that Bob has invested $1500 in the savings account, $3000 in the CD, and $5500 in the money market fund.

(35) 8 6

4 1AB

⎡ ⎤= ⎢ ⎥−⎣ ⎦

(36) Letting l be the length of the rectangle and w be the width we

have 120lw = and 2 2 46l w+ = . Substituting 120

l for w in the

second equation gives us 120

2 2 46ll

⎛ ⎞+ =⎜ ⎟⎝ ⎠

. We multiply through by

l and solve the resulting quadratic equation, giving us a length of 15 in. (and a width of 8 in.). The diagonal thus has length

2 28 15+ in, or 17 in.

(37) ( ) ( )11 2 1nna n−= − ⋅ −

(38) ( )15

2

0

2k

k=

+∑

(39) ( )17 4 1 4 21na n n= − + − = − (40) 630 seats

(41) 41

42

218

3a ⎛ ⎞= ⋅ ⎜ ⎟

⎝ ⎠

(42) 18

5421 13

aS

r= = =

− −

Revised 12/20/04

(43) Before simplifying, we get

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )4 0 3 1 2 2 1 3 0 44 4 4 4 43 5 3 5 3 5 3 5 3 5

0 1 2 3 4x x x x x

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + − + − + − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Simplifying, this becomes 4 3 281 540 1350 1500 625x x x x− + − + .