Download Topics for Review:

Math 4
Name_______________________
Review Problems
Topics for Review #1
Functions
function concept [section 1.2 of textbook]
function representations: graph, table, f(x) formula
domain and range
Vertical Line Test (for whether a graph is a function)
evaluating functions and solving equations using graphs (on paper and on calculator)
function properties [1.2]
increasing, decreasing, constant
local maxima and minima, absolute maxima and minima
even and odd symmetry (recognizing graphically and confirming algebraically)
drawing graphs with specified properties
piecewise-defined functions [1.3]
function operations [1.4]
addition, subtraction, multiplication, and division
composition
inverse functions [1.5]
finding an inverse by reversing the steps of a rule
Horizontal Line Test (for whether a graphed function has an inverse)
finding an inverse by exchanging the x and y variables of a graph or table
finding an inverse graphically by reflecting across the line y = x
finding an inverse algebraically
transformations, graphs and functions: translations, dilations and reflections [1.6]
Polynomials
linear functions and quadratic functions in various forms [2.1]
vertex of a quadratic function (including completing-the-square into vertex form) [2.1]
graphs of polynomial function (end behavior, multiplicity of zeros) [2.3]
division concept (relationships between dividend, divisor, quotient and remainder) [2.4]
polynomial algebraic calculations (especially factoring and dividing) [2.4]
Remainder Theorem and Factor Theorem (proving and using) [2.4]
finding zeros and factorizations of polynomials [2.3 for real, 2.5 for complex]
using theorems about complex zeros: Fundamental Theorem of Algebra, Complex Conjugates
Theorem, odd degree theorem [2.5]
application/modeling problems [in various sections, and more on handout problem sets]
Rational Functions
Graphs of rational functions, including all critical features – asymptotes, holes and general shape
[2.6]
Limits to describe end behavior and behavior at discontinuities [2.6]
Equation solving [2.7]
Application problems [2.7]
Sign analysis and sign charts [2.8]
Math 4
Name_______________________
Review Problems
Problems
1. Consider the following functions:
f x  3 tan x  2 sin x
g ( x) = x
h ( x) = x 2 + 5
a. Find ( h o g) ( x ) and ( g o h) ( x ) and their domains.
b. Is f(x) even or odd? Prove it.
2. Describe a sequence of transformations to turn the graph of f ( x ) into the graph of g ( x ) .
a. f ( x ) = 2x 2 +1, g ( x ) = -2 ( x + 7) + 5
2
æxö
b. f ( x ) = sin x, g ( x ) = 3sin ç ÷ - 4
è2ø
3. Suppose g ( x ) =
2x - 9
.
x-5
a. Find g-1 ( x ) .
b. Write g(x) in the form: g(x) =
a
+k
x -h
1
c. Describe the transformation of g(x) from f (x) = .
x
d. Describe the zeros and discontinuities(s) of g(x).
e. What are the domain and range of g(x)?
f. Sketch a graph of g(x).
4. Find a cubic polynomial, C(x), with real coefficients has the following properties:
1 2i is a complex root.
the graph of C(x) has an x-intercept at x = 2 .
C(1) = -12
Math 4
Name_______________________
Review Problems
5. All of the zeros are visible on the graph of this polynomial function, and there are no
multiplicities higher than 2.
a. Find a function formula for this polynomial graph.
b. Without doing the division, find the remainder when this polynomial is divided by (x + 2)
6. Solve these equations. Find all solutions.
a. x 3  x 2  23x  42
b. sin q = -
3
2
7. Consider the following function:
g ( x) =
2(x 2 - 5x -14)
x2 - 4
a. Find the discontinuities of g(x) and identify whether each is an asymptote or a hole.
b. Write limit statements describing the end behavior of g(x).
c. Write limit statements describing the discontinuities of g(x).
d. With the help of your calculator, sketch a graph of g(x).
Math 4
Name_______________________
Review Problems
Topics for Review #2
Exponential and logarithmic functions
Exponential functions and their graphs [3.1]
Modeling growth and decay using exponential functions [3.2]
Meaning of logarithms, including exponential and logarithmic functions as inverses [3.3]
Logarithmic functions and their graphs [3.3]
Solving exponential equations using logarithms [3.5]
Arithmetic properties of logarithms (3 properties proved from 3 exponent properties) [3.4]
Change-of-base for exponentials and logarithms [3.4]
Trigonometric Functions
Measures of angles, arcs, and sectors [4.1]
Definitions of sin, cos, tan, cot, sec, csc
right triangle definition for acute angles [4.2]
circular definition for all angles [4.3]
Trigonometric functions and their graphs
sinusoids (sin and cos, possibly with translations and dilations) [4.4]
tan, cot, sec, and csc [4.5]
inverse trigonometric functions [4.7]
Finding trigonometric function values, exactly or approximately [4.2-4.3]
Solving trigonometric equations [4.4, 2/11 handout]
Sinusoidal modeling [4.4, 4.8, 2/12 handout]
amplitude, vertical shift, period, and phase shift
real-world applications
Geometric applications [4.2, 4.8, 2/12-2/13 handouts]
The Law of Sines (including ambiguous case)
The Law Cosines
Problems
8. a. Write a function formula for a sinusoidal function f(x) having the following properties:
Two consecutive maximum points of f(x) are located at (3, 5) and (7, 5).
The graph of f(x) is tangent to the x-axis.
b. Suppose that the graph of g(x) is formed by shrinking f(x) horizontally by a factor of 8.
Write a function formula for g(x).
9. For triangle RST, given that R = 45, RS = 8, and ST = 10, find the remaining
measurements of the triangle. There may be one or two sets of answers.
Math 4
Name_______________________
Review Problems
10. For quadrilateral WXYZ, given that XY = 10, Y = 50, YZ = 8, Z = 110, and ZW = 6:
a. Find the remaining sides and angles.
b. Find the area.
11. Suppose that the depth of the water in a harbor is 20 feet at low tide, 30 feet at high tide, and
fluctuates in such a way that it can be modeled with a sinusoidal function.
Let t represent time measured in hours, and let D(t) represent the water’s depth in feet at time t.
Suppose that a low tide occurs at time t = 5.4 and the next high tide occurs at time t = 11.6.
a. Using the sine function, write a function formula for D(t).
b. Find all times t at which the depth of the water is 28 feet. (Use decimal approximations
accurate to the nearest 0.01.)
12. The half-life of radium is about 1600 years. Suppose that an archeological sample is found to
contain 200 milligrams of radium. How many milligrams of radium were there 5,000 years ago?
13. Let a = log3 2 and b = log3 5. Express log3 450 in terms of a and/or b.
æ bx ö
14. Given w = ç 3 ÷ , write logb w in terms of logb x and logb y .
èy ø
1
15. Prove that log b c 
. (You can use any formulas proved in class.)
log c b
2
æ 2ö
16. Find an equation for an exponential function passing through the points ç -1, ÷ and ( 2,18) .
è 3ø
17. An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person’s
system decreases by about 29%. How much ibuprofen is left after 6 hours? How long will it take
for the amount of ibuprofen to decrease to 10% of its original value?
Math 4
Name_______________________
Review Problems
Topics for Review #3
Trigonometric Identities [5.1-5.2]
angle sum(difference) identities
double angle identities
solving and proving trigonometric equations with identities
Vectors [6.1-6.2]
vector concept: run-and-rise or direction-and-magnitude
equivalent arrows representing the same vector
vector operations (addition, subtraction, scalar multiplication, dot product)
using both algebraic and visual approaches
vector magnitude (length) and unit vectors
parallel and perpendicular vectors; angle between vectors
geometric proofs using vectors
Parametric equations [6.3]
parametric equation graphing, by hand and on calculator
writing parametric equations for lines, line segments, and circles
eliminating the parameter t
modeling 2-D motion using parametric equations (incl. gravity problems)
Polar coordinates [6.4]
polar coordinate concept
converting coordinates between rectangular and polar
converting equations between rectangular and polar
polar equation graphing, by hand and on calculator
application problems involving polar graphing and distances
Complex numbers in polar form [6.6]
writing complex numbers in polar form (trigonometric form)
polar form multiplication, division, and powers (DeMoivre’s Theorem)
Problems
1
1

 2 sec 2 x . Present your proof as a chain of
1  sin x 1  sin x
1
1

equal expressions that begins with
and ends with 2 sec 2 x .
1  sin x 1  sin x
18. Prove the identity
19. Derive a formula for cos(2u) that involves sin2 u but does not involve cos2 u.
æp
ö
20. Prove that cos(q ) = sin ç - q ÷ using sin(α – β).
è2
ø
21. A woman has a collection of bills in her wallet. Each bill is worth $1, $5 or $10. She has 19
bills, amounting to $92. There are twice as many ones as tens. What bills will you find in her
wallet?
Math 4
Name_______________________
Review Problems
22. At Fenway Park, the Green Monster is 37 feet high and is located 310 feet from home plate.
Suppose a batter hits a ball at an elevation of 4 feet from the ground at an initial velocity of
112 ft/s at an angle of 28º. Will a spectator seated atop the Green Monster be able to catch
the ball?
23. Vector v is the directed line segment
from (-7,1) to (5,-8). Vector w has
component form <2,5>.
a. Draw the vector w + .5v on the axes
at right.
b. Use a vector method to find the
angle between w and v.
c. Find a vector of magnitude 2 in the
direction of v.
24. Let z = −1− 2i .
a. Find z 8 . Express you answer in polar form
b. Find z .
25. Fill in the conversion table below:
cartesian
convert
polar
1 3
(- , )
2 2
(-4, 56p )
x 2 + y 2 = 12
r = 4 cscq
26. a. Write a polar equation (in r = L form, if possible) for the line whose rectangular
equation is y = 2x + 3.
b. Convert polar equation r =
1
into a rectangular equation (in y =L form,
cos( )  sin(  )
if possible).
c. Convert the polar equation r  4 cos to rectangular form.