Pre-Calculus
Review – 1st semester exam
Name ____________________________
**The final exam covers chapters P,1,2,3 and right triangle trigonometry and will account
for 20% of your final grade.**
Chapter P: (pg.24,#40,42; pg.35, #27,28; pg. 45,#14,19,21,56; pg. 53, #1,3,5,15,35)
For questions 1 and 2, solve the inequality and draw a number line graph of the solution set.
1) −1 ≤ 3 x − 2 < 7
2) 4 (1 − x ) + 5 (1 + x ) > 3 x − 1
For questions 3 and 4, find a slope-intercept form equation for the line.
3) The line through the points (-4,5) and (4,3)
4) The line through the points (4,2) and (-3,1)
5) Solve using the quadratic equation: 2 x 2 − 3 x + 1 = 0
6) Solve by factoring: x 2 − x − 20 = 0
7) Solve graphically: 4 x 2 − 8 x + 3 = 0
8) Solve algebraically:
3x
1
7
+
= 2
x + 5 x − 2 x + 3 x − 10
For questions 9 – 13, Solve algebraically and express your answer in interval notation:
9) x + 4 ≥ 5
10) x − 3 < 2
11) 4 − 3 x − 2 < 4
12) x 3 − x ≥ 0
13)
x+2
≥1
2x − 3
Chapter 1: (pg. 96, #25,34,41,45,47; pg. 122, #9,11; pg. 133, #7,41,43,45,47;
pg. 150, #14,16,21,28 (w/ h- and v-asymptotes, 34,37, 41,42,53,54,57)
1) Use a grapher to find all local maxima and minima and the values of x where they occur. Round
values to 2 decimals places. h( x) = − x3 + 2 x − 3
2) Graph the function and identify intervals on which the function is increasing, decreasing, or
constant. Remember, your intervals are based on the “x” values.
f ( x) = x3 − x 2 − 2 x
3) State whether the function is odd, even, or neither. Support graphically. f ( x) = 2 x 4
4) State whether the function is odd, even, or neither. Support graphically. g ( x) = 2 x 3 − 3 x
5) Use a method of your choice to find all h- and v- asymptotes. (You should be able to do it
x+2
x2 + 2
and f ( x) = 2
both graphically and algebraically.) g ( x) =
3− x
x −1
6) Find ( f o g )(3) and ( g o f )(−2) if f ( x) = 2 x − 3 and g ( x) = x + 1
7) Find f ( g ( x)) and g ( f ( x)) if f ( x) = 3 x + 2 and g ( x) = x − 1
8) List, in order, the transformations applied to y = x 2 to obtain the graph of y = ( x − 1) + 3 .
2
For 9 and 10, describe a basic graph (original) and a sequence of transformations that can be
used to produce a graph of the given function.
9) y = 2 ( x − 3) − 4
2
10) y = ( 3 x ) − 4
2
For 11 and 12, write the equation of the new function based on the given transformations
applied to the given function.
11) y = x 2 ; Vertical stretch by a factor of 3; shift right 4 units.
12) y = x ; shift left 2 units; vertical stretch by factor of 2; shift down 4 units
For 13 and 14, find the domain and range of the function. You should be able to do it
algebraically and graphically.
13) h( x) = ( x − 2 ) + 5
2
14) k ( x) = 4 − x 2 − 2
15) Find all h- and v-asymptotes for the function: f ( x) =
5
(Algebraically, then graphically.)
x − 5x
2
16) State all intervals on which the function is increasing. (Remember, the intervals are based
off the “x” values of the ordered pairs.)
y=
x2 − 1
x2 − 4
17) Use a grapher to find all relative maxima and minima. Also state the value of x at which each
relative extrema occurs.
y = x3 − x
18) Graph the function and state whether it is even , odd, or neither.
For 19 and 20, find the equation for f −1 ( x) .
19) f ( x) = 2 x + 3
20) f ( x) = 3 x − 8
y = 3x 2 − 4 x
For 21 and 22, let f ( x) = x and g ( x) = x 2 − 4 .
21) Find an expression for ( f o g )( x) and give its domain.
22) Find an expression for ( g o f )( x) and give its domain.
23) Describe the end behavior of the function y = x using limit notation.
Chapter 2 (pg. 170, #29; pg. 196, #29,30; pg. 208, #3,7,13,17,19,21;
pg. 236, #17,19,25,27,33,35,43; pg. 246, #1,7,13,29,55)
1) Sketch a graph of the function by hand. f ( x) = x 2 − 4 x + 6
2) Describe the end behavior of the following graphs. (Use correct limit notation)
a) f ( x) = 3 x 4 − 5 x 2 + 3
b) f ( x) = − x 3 + 7 x 2 − 4 x + 3
3) Divide f(x) by d(x). f ( x) = x3 + 4 x 2 + 7 x − 9; d ( x) = x + 3
4) Divide using synthetic division.
x3 − 5 x2 + 3x − 2
x +1
5) Use the remainder theorem to find the remainder when f(x) is divided by x – k.
f ( x) = 2 x 2 − 3 x + 1; k = 2
6) Use the remainder theorem to find the remainder when f(x) is divided by x – k.
f ( x) = 2 x 3 − 3 x 2 + 4 x − 7; k = 2
7) Use the factor theorem to determine whether the first polynomial is a factor of the 2nd.
x − 1; x3 − x 2 + x − 1
8) Use the factor theorem to determine whether the first polynomial is a factor of the 2nd
x − 2; x3 + 3 x − 4
9) (pg. 236 #17, 19 – use book for graph)
10) Find the asymptotes and intercepts of the function. f ( x) =
2
x −3
11) Find the asymptotes and intercepts of the function. f ( x) =
x−2
x − 2x − 3
2
12) Find the asymptotes and intercepts of the function. f ( x) =
2x2 + x − 2
x2 − 1
x2 − 2x + 3
13) Find the asymptotes and intercepts of the function. f ( x) =
x+2
14) Solve the equation algebraically. Check for extraneous solutions!
3x
1
7
+
= 2
x + 5 x − 2 x + 3 x − 10
15) Determine the values of x that cause the polynomial function to be a) zero, b) positive, and
c) negative f ( x) = ( x + 2 )( x + 1)( x − 5 )
16) Complete a sign chart and solve the polynomial. Express your answer in interval notation.
( x + 1)( x − 3)
2
>0
17) Solve the polynomial inequality graphically. Express your answer in interval notation.
x3 − x 2 − 2 x ≥ 0
18) Solve the inequality using a sign chart. Support your answer graphically. Express your answer
x −1
<0
in interval notation.
x2 − 4
19) Designing a cardboard box. See pg. 247, #55.
Chapter 3: (pg. 270, #15,22,31,33; pg. 279, #1,3,11,29,33,39; pg. 291, #1,11,13,29,37,39; pg.
299, #1,5,15,25,31,35; pg. 313, #1,5,13; pg. 324, #5,7,27)
1) and 2) Describe how to transform the graph of f into the graph of g. Sketch the graph by
hand and support your answer with a grapher.
1) f ( x) = 2 x ; g ( x) = 2 x −3
2) f ( x) = e x ; g ( x) = −e −3 x
3) and 4) State whether the function is exponential growth or decay and describe its end
behavior using the correct limit notation.
3) f ( x) = 3−2 x
4) f ( x) = .5 x
5) and 6) Tell whether the function is exponential growth or decay and find the constant
percentage rate of growth or decay for the function.
5) f ( x) = 3.5 • 1.09 x
6) f ( x) = 78963 • .968 x
7) Find the exponential function that satisfies the given conditions.
Initial population = 502,000; Increasing at a rate of 1.7% per year.
8) The population of Knoxville is 475,000 and is increasing at the rate of 3.75% each year.
Predict when the population will be 1 million. (Set up an equation and solve graphically.)
9) The half-life of a certain radio-active substance is 14 days. There are 6.6 grams presently.
a) Express the amount of substance remaining as a function of time t.
b) When will there be less than 1 gram remaining? (set up an inequality and solve
graphically)
10) The number B of bacteria in a petri dish after t hours is given by B = 100e.693t . When will the
number of bacteria be 200? Estimate the doubling time of the bacteria.
For 11 – 13, evaluate without a calculator.
11) log 4 4
13) ln e3
12) log 2 32
For 14 – 16, Solve by changing to exponential form. Then use calculator to approximate solution.
15) ln x = 2.8
14) ln x = −2
16) log x = 8.23
For 17 and 18, Evaluate using a calculator.
17) log 2 7
18) log.5 12
19) Graph the function. Then state its domain and range. f ( x) = log 4 x
20) Use properties of logs to write the expression as a sum or difference of logs.
log
3
x
x2
21) Use properties of logs to write the expression as a sum or difference of logs. ln 3
y
22) Use properties of logs to express as a single logarithm. log x + log y
23) Solve algebraically. 1.06 x = 4.1
(You may use a calculator to approximate your solution.)
24) Solve algebraically. 50e.035 x = 200 (You may use a calculator to approximate your solution.)
25) Solve algebraically. log 4 ( x − 5 ) = −1 (You may use a calculator to approximate your solution.)
26) Find the amount A accumulated after investing a principal P for t years at interest rate r
compounded k times per year.
a) P = $1500; r = 7%; t = 5, k = 4
b) P = $40,500; r = 3.8%; t = 20; k = 12
27) Determine how much time is needed for an investment to double in value if interest is
earned at the rate of 5.75% compounded quarterly.