Conics
a) Identify the conic section. Re-write each equation in standard form and graph the conic
section.
b) If it is a parabola, give its vertex, focus and directrix.
c) If it is an ellipse, give its center, vertices, and foci.
d) If it is a hyperbola give its center, vertices, foci, and equations for asymptotes.
31)
a. Conic section________________________
e) Focus or Foci ____________________________
b. Equation____________________________
c. Center_____________________________
d. Vertex or vertices____________________
4x 2 + y 2 - 8x + 4y - 8 = 0
f) asymptotes ______________________________
a) Identify the conic section. Re-write each equation in standard form and graph the conic
section.
b) If it is a parabola, give its vertex, focus and directrix.
c) If it is an ellipse, give its center, vertices, and foci.
d) If it is a hyperbola give its center, vertices, foci, and equations for asymptotes.
31)
a. Conic section________________________
e) Focus or Foci ____________________________
b. Equation________________________________
c. Center_____________________________
d. Vertex or vertices____________________
4x2 – 9y2 + 24x – 36y – 36 = 0
f) asymptotes ______________________________
Review - Conic Sections
PART I: SHOW ALL WORK (ON A SEPARATE SHEET OF PAPER):
For 1 – 5: Write the standard form equation of each conic section using the given
information:
(1)
A parabola whose focus is at (2, 4) and equation of the directrix is x = 6.
(2)
A hyperbola whose center is at (2, 4), length of vertical transverse axis is 24, and
length of conjugate axis is 18.
(3)
An ellipse whose center is at (-2, -4), length of vertical major axis is 24, and length
of minor axis is 18.
(4)
A hyperbola whose center is at the origin, length of horizontal conjugate axis is 8,
and distance of foci from center is 5.
(5)
An ellipse whose center is at the origin, length of horizontal major axis is 16, and
distance between foci is 12.
For 6 – 12: Given the general form equation:
(a) identify the conic section represented
(b) write the equation in standard form
(c) find the necessary coordinates and equations relevant to that conic section
(d) graph the equation (on graph paper)
(6)
x2 – 4y2 – 4x – 24y – 36 = 0
(7)
x2 + 8x + 8y = 0
(8)
x2 + y2 + 12x + 10y + 45 = 0
(9)
16x2 + 4y2 + 96x – 16y + 96 = 0
(10) 16x2 + 9y2 – 192x – 72y + 576 = 0
(11) 25y2 – 9x2 – 225 = 0
(12) y2 – 8x – 4y – 20 = 0
Trigonometry
Verify the trigonometric identity.
cos x
cos x

 2 sec x
1  sin x 1  sin x
Verify the trigonometric identity.
1
 cot x  tan x
sin x cos x
Precalculus
Trigonometry Review
Final Review
Reciprocal Identities
1
Ratio Identities
𝑠𝑖𝑛Ө
sin2Ө + cos2Ө = 1
𝑐𝑜𝑠Ө
tan2Ө +1 = sec2Ө
secӨ=𝑐𝑜𝑠Ө
tanӨ =𝑐𝑜𝑠Ө
1
cotӨ= 𝑠𝑖𝑛Ө
cscӨ = 𝑠𝑖𝑛Ө
Pythagorean Identities
1
1+ cot2Ө = csc2Ө
cotӨ = 𝑡𝑎𝑛Ө
Summary of Cofunctions
Sine and Cosine
sinӨ=cos(90-Ө)
cosӨ=sin(90-Ө)
Tangent and Cotangent
tanӨ=cot(90-Ө)
cotӨ=tan(90-Ө)
Secant and Cosecant
secӨ=csc(90-Ө)
cscӨ=sec(90-Ө)
COFUNCTIONS
COTERMIANL ANGLES AND REFERENCE ANGLES
TRIGONOMETRIC EQUATIONS
1) Substitute equal values when necessary (usually for double angle, more than one term
in the same equation…) – your goal is to get the original equation written in terms of
one trigonometric function if possible
2) Once all of the substitutions are made, solve as if the trig function is a variable… I.e.
sin=y, cos =x… this allows for easier manipulation
3) When the trig functions are solved for values, solve for the theta Ө. Most equations have
two theta values that will satisfy---
** so if the original is quadratic you may potentially have 4
solutions
Identities
Verifying an identity1) Start with the most complex side of the equation
2) Substitute in equal values if necessary
3) DO NOT move terms to the other side of the equal sign
4) Simplify using all valid algebraic manipulations
Graphs of Trig Functions
y = a sin b(x+c)+k
y= a cos b(x+c)+k
a is amplitude
vertical shift
b is frequency
y = sin x
c is the phase shift
k is the
Different Amplitude--y= 2 sin x
y= cos x
y = 3 cos x
y= tan x
y = 2 tan x
Changes in frequency
y= sin 2x
Vertical shifts
y = sin 2x + 3
y= cos 3x
Phase shifts
𝜋
y = sin 2( x+ 2 )
1
y = cos 2 x – 2
Mixed:
Graph:
𝜋
y = 2 sin 2(x- 2 ) -2
𝜋
y= cos 3( x – 3 )
Polynomial Functions
Solve the equation over the set of real and complex numbers.
g(x) = x4 – 2x3 –7x2 + 18x - 18
a) Using Descartes’ Rule of Signs determine the possible number of positive and negative real
zeros.
Positive real zeros_______________________ Negative real zeros_______________________
b) Using the Rational Root Theorem to list all the potential rational zeros.
List of potential rational zeros ___________________________________________________
c)
a) Write the function as the product of linear factors and not a factorable quadratic
function.
b) List all the real and complex zeros.
Linear factorization____________________________________________________
Zeros_______________________________________________________________
Given the function f ( x)  3x 4  x 3 13x 2  5x 10
a) Using Descartes’ Rule of Signs determine the possible number of positive and negative real
zeros.
Positive Zeros______________________________________
Negative Zeros_____________________________________
b) Using the Rational Root Theorem to list all the potential rational zeros.
Potential Zeros:_____________________________________________________________
c) Write the polynomial as the product of factors. Do not factor any irreducible factors.
d) List all complex and real roots.
Real Roots_________________________ Complex Roots__________________________
Rational Functions
Use R(x) to answer the questions. r ( x) 
x2  9
x 2  2x  3
a) Find the domain
b) Find the x and y-intercepts, if any
c)
d)
Find the vertical asymptote(s) and/or any holes , if
any,
Find the horizontal or oblique asymptote if any,
e. Sketch the graph of the rational
function using the information
above. Label the graph
appropriately.
Use R(x) to answer the questions. r ( x) 
a) Find the domain
x 2  3x
x2  x  6
b)
Find the x and y-intercepts, if any.
c)
d)
Find the vertical asymptote(s) and/or any holes , if
any,
Find the horizontal or oblique asymptote if any,
a) Sketch the graph of the rational function using the information above. Label the graph
appropriately.
Mathematical Induction
Use Mathematical Induction to prove true for all natural numbers.
1 + 5 + 9 + … + (4n - 3) = n (2n-1)
Using the Principle of Mathematical Induction, show the following is true for all natural numbers
n:
1 + 3 +32 + …+ 3n-1 =
1 n
(3  1)
2
Using the Principle of Mathematical Induction , show the following is true for all natural
numbers n:
Piecewise Functions
Graphing a Piecewise Function
f(x)
x 2 , ........ ..x  1
= 2....... .. ..x  0
 x  2......if x  0

a) Graph f
b) Find f(-2),
f(0), and f(3)
c) Determine the
domain of f
d) Use the graph
to find the range
of f
e) Is f continuous
on its domain?
Write the piecewise function.


f (x)  


b)


f (x)  


Exponential and Logarithm Functions
Solving Exponential and Logarithmic Equations
a) 2 x  32
b) log 10 x  1
c) log 3 (4 x  7)  2
e) log 4 ( x  3)  log 4 (2  x)  1
g) 5 x  2  33x  2
d) 2 log 5 x  log 5 9
f) 3 x  1  81
h) e x +
3
= πx
Identify the domain and the range for the given logarithm functions:
a) f(x) = log2(1 – x)
b) g(x) = log5(
1 x
)
1 x
c) y  log 4 (3x  3)  2
If you invest $5000 in a stock that is increasing in value at the rate of 3% per year, then the value
of your stock is given by the function f(x) = 5000(1.03) x , where x is measured in years.
a) Assuming that the value of your stock continues growing at this rate, how much will your
investment be worth in 4 years?
b) When will your investment be worth $8000?
Sequences and Series
1) Find the 78th term of the arithmetic sequence
2, 6, 10, 14, 18, …..
2) Find the 1st term of an arithmetic sequence, the common difference, and find the
recursion formula when the 12th term is 30 and the 22nd term is 50.
3) Determine whether the given sequence is arithmetic, geometric, or neither. If the
sequence is arithmetic, find the common difference; if it is geometric, find the
common ratio.
n
2
b) {   }
3
a) {2n - 5}
4) Find the 22nd term of the geometric sequence
5) Find the sum
3 3 2 33
315

  ... 
9 9
9
9
4, 8, 16, ……
6) Find the sum
3 + 0.3 + 0.03 + 0.003 + ….
7) The corner section of a football stadium has 15 seats in the first row and 40 rows in
all. Each successive row contains two additional seats. How many seats are in this
section?
8) A certain rubber ball when dropped bounces in such a way that the height of the first
bounce is 80% of the initial height, and height of each bounce after that is 80% of the
height of the previous bounce.
a) Find the height of the fifth bounce if the ball is dropped from an initial height of 20
feet.
b) Find a formula for the height of the nth bounce if the ball is dropped from an initial
height of h feet.