# Math Final Review

Highlights from Chapters 1 & 2:

Absolute value

Simplification:

exponent rules (cf log rules )

taking roots

(or
simplifying any even power)

there are still two solutions to the problem

However, it is wrong to say consider x = −8

it is painfully wrong to say

unless you know one of a or b is equal to 0.

combining rational expressions (getting common denominators )

compound fractions (simplifying fractions within fractions)

rationalizing denominators and numerators

Factoring:

difference of squares x^2 − y^2 = (x + y)(x − y)

sum of cubes x^3 + y^3 = (x + y)(x^2 − xy + y^2)

difference of cubes x^3 − y^3 = (x − y)(x^2 + xy + y^2)

quadratic equation, solutions to ax^2 + bx + c = 0 for non-zero a are:

completing the square

get 0 on one side and factor

if f(x)h(x) = 0 then f(x) = 0 or h(x) = 0

**this is not true with vectors**

equations of quadratic type

extraneous roots:

may be created when

multiplying both sides of an equation to clear denominators

raising both sides of an equation to an even exponent

clearing logs with exponentials

if you do any of these things you must check your answers

Inequalities:

interval notation sign diagrams

properties p. 85

Distance between point
is

Midpoint formula for P _{1} and P_{2} is

Graphs of equations:

find your x and y intercepts, plot them especially when they are nice

x-intercept = when your function hits the x-axis = root = zero = when y is zero

y-intercept = when your function hist the y-axis = when x is zero

symmetry with respect to x-axis, y-axis, origin

you may want to think about this for the conics which we have recently been
discussing.

Lines:

slope

perpendicular, parallel

Functions

Definition: a function f assigns to each x in its domain a unique y in its range

(each x gets assigned to one, and only one y value)

evaluation

graph (this is where you might apply the vertical line test)

domain, range

Graphs of Functions:

even f(−x) = f(x)

odd f(−x) = −f(x)

how does this correspond to symmetry

shifting, vertical and horizontal stretches, reflecting

Introductory parabola material

How does this compare to our new view of parabloas?

Standard form

Vertex (max/min)

add, multiply, divide

how does this change the domain?

Composition of functions

domain

**Chapter 3**

3.1

Intermediate Value Theorem: If f is a polynomial function and f(a) ≠ f(b) for a≤b,
then f takes

on every value between f(a) and f(b) in the interval [a, b].

Sketch the graph of a function from its sign chart.

Find the sign chart from the (factored) equation of a function.

3.2 Polynomial Long division

Division algorithm : If f(x) and p(x) are polynomials and if p(x)
≠ 0, then
there exist unique

polynomials q(x) and r(x) such that f(x) = p(x) · q(x) + r(x). Where r(x) = 0 or
the degree of r(x)

is less than the degree of p(x).

Remainder Thm: If a polynomial f(x) is divided by x − c, then the remainder is f(c).

Factor Thm: A polynomial f(x) has a factor x − c if and only if f(c) = 0.

3.5 Rational Functions

domain

vertical asymptotes

horizontal asymptote

oblique asymptote

hole

sketching the graph

finding an equation given the graph

3.6 Variation

direct (write xαy for “x is directly proportional to
y ”)

inverse

Chapter 4

4.1 Inverse Functions

Definition of 1-1

Horizontal line test

Theorem on Inverse Functions: Let f be a one-to-one function with domain D and
range R. If g is

a function with domain R and range D, then g is the inverse function of f if and
only if both of the

following conditions are true: (1) g(f(x)) = x for every x in D, and (2) f(g(y))
= y for every y in R.

domain of f^{−1} = range of f, and range of f^{−1} = domain of f

graph of f^{−1} is a reflection of the graph of f through the line y = x

4.2 Exponential functions:

if the base is greater than 1, the graph is increasing

if the base is less than 1 (but greater than 0), the graph is decreasing

either way the y intercept is 1 (a^{0} = 1 for any base a)

Note: we do not consider 0, 1, or negative numbers as bases.

Exponential funcitons are 1-1

Solve exponential equations using 1-1

Sketch graphs of exponentials, shifting, multiplying by a constant

compound interest

4.3 Natural Exponential Function

continuously compounded interest

growth/decay

sketch graph

finding zeros

4.4 Logarithms

y = log_{a}x if and only if x = a^{y}

Note: this means that log_{a}u is the inverse function of a^{y}

converting between logs and exponentials

evaluating logs

solving log equations

sketching graphs

common log is log_{10}

natural log is log_{e}(denoted ln)

4.5

Properties of Logarithms:

proofs of above properties (using the definition and corresponding properties
for exponents)

Note: there is not a convenient way to break up log_{a}(u + v)

applying these rules solving equations with logs more graphing

4.6 Change of base formulas

Chapter 5

5.1 Angles

initial side, terminal side, vertex

coterminal angles

standard position

counterclockwise = positive

clockwise = negative

quadrants

finding coterminal, complementary, supplementary angles

acute, obtuse

radians (convert from degrees to radians and radians to degrees

180º = π
radians

length of a circular arc: s = rθ

area of a circular sector:

5.2

Trig functions

reciprocal identities
etc.

given one trig function, find the values of the others

trig functions for 30º
, 40º
and 60º (chart on p 375)

expressing one trig function in terms of another one

showing an equation is an identity

5.3

extend trig functions to all of the unit circle, not just triangles

signs of each trig function in the different quadrants

trig functions of multiples of π/2

range, domain, even/odd symmetry, y-intercept, x-intercepts

graphs (these are all periodic functions)

using a graph to find solutions to an equation

5.4

reference angles

exact values

5.5 & 5.6

graphing: change period, amplitude, phase shift, vertical shift

if y = a sin(bx + c) + d the amplitude is |a| the period is
and the phase shift is −c/b

finding, amplitutde, phase shift, period from a function

cosine, tangent, cotangent, secant and cosecant as well

5.7

given some information about a right triangle, solve for the rest of the
information.

Chapter 6

6.1 Verifying Trig Identities

you may work with both sides of an equation, however you may not multiply both
sides by a variable

expression.

recall that to show something is not an identity, it is enough to find one
number for which the two

sides are unequal.

6.2 Trigonometric Equations

using multiple angle formulas

set equal to 0, factor

find all solutions

find solutions in an interval

6.4

Double Angle Formulas:

Note: this should help you remember the sign of cos(u + v) because it is not
true that cos(2u) = 1

for all u.

Half-Angle Identities:

Half-Angle Formulas:

also for tangent:

6.5

Product to sum formulas:

Sum to product formualas:

6.6 Inverse trig functions

summary of relationships between f and f^{−1} on p510

y = arcsin(x) if and only if x = sin(y) for

Domain: Range:

so sin(arcsin(x)) = x

if x is outside this range, this is undefined

also arcsin(sin(y)) = y if

if y is outside this range it is never a value of arcsin (arcsin(sin(y)) is still defined)

y = arccos(x) if and only if x = cos(y) for

y = arctan(x) if and only if x = tan(y) for any real number x and

note the domain must be restricted in order to define the inverse functions.

evaluating (sum forumalas)

changing into an algebraic expression

solving equations

finding an inverse function

Chapter 7

7.1 Law of Sines

if ABC is a triangle laveled in the usual manner then

7.2 Law of Cosines

The area of a triangle is one-half the product of the lengths of any two
sides multiplied by the sine

of the angle between them.

Heron’s Formula: the area A of a triangle with sides a, b, and c is given by:

where s is one-half the perimeter:

7.3 Vectors

directed line segment (inital and final points)

magnitude (length of that line segment)

direction (angle, initial to final)

addition: place initial point of one vector on the final point of the other.

doing this in two ways creates a parolleogram whose diagonal is the vector
representing the sum

a position vector

inital point at origin, final point at

magnitude is therefore:

addition:

multiplication by a scalar: the zero vector is

this is the same as multiplying by −1

properties of vectors: see page 555

Definition of i (not to be confused with imaginary i) and j

unit vectors have magnitude 1

horizontal component of

vertical component of

7.4 The Dot Product

this is a real number, not a vector

properties of dot product p. 566

vectors are parallel if the angle between them is 0 or π

they are orthogonal if the angle between them is π/2

so if θ is the angle between two nonzero vectors a and b then

two vectors are orthogonal if and only if a · b = 0

Chater 8

8.1 & 8.2 Systems of equations

If you have two equations in two variables (say x and y), to use the method of
substitution :

-solve for one variable (for example x) in terms of the other (here y)

you now have something that reads x = f(y) where f(y) is just some expression
involving y.

-substitute that expression in your other equation

you should now have an equation involving only the one variable (y in our
example)

-solve that equation (for y)

-for each value of y produced, use the original expression (x = f(y)) to get a
value for x

-check each pair of solutions (x, y)

if you have more variables and more equations, it is possible to modify this
process

If you have a system of linear equations you may want to change it into an
equivalent system by:

-interchanging two equations

-multiplying an equation by a non-zero constant

(equivalently dividing by a non-zero constant)

-adding a constant multiple of one equation to another

in this case you must multiply each term of the equation (on both sides of =) by
the same constant

The types of systems of linear equations we discussed are:

Nonparallel lines = one solution = “consistent system”

Identical lines = infinite number of solutions = “dependent and consistent
system”

Note: this does not mean all pairs (x,y) satisfy the system

Parallel lines = No solution = “inconsistent system”

Chapter 10

10.1 Parabolas

introduction to conic sections

a parabola is the set of points in the plane equidistant from a fixed point
(focus) and a fixed line

(directrix)

the vertex is the point halfway between the focus and directrix

the axis is the line running through the vertex and focus

Note: a parabola is symmetric with respect to its axis

If the axis of a parabola is vertical, the equation is:

(x − h)^2 = 4p(y − k) where the vertex is (h, k) the focus is (h, k + p) the
directrix is y = k − p

if p is positive the parabola opens up, if p is negative it opens down.

If the axis of a parabola is horizontal, the equation is:

(y − k)^2 = 4p(x − h) where the vertex is (h, k) the focus is (h + p, k) the
directrix is y = h − p

if p is positive the parabola opens right, if p is negative it opens left.

you should be able to go between graphs, equations, focus/vertex/directrix

that is, given some little information you can produce the rest

10.2 Ellipses

an Ellipse: the set of points (P), the sum of whose distances from two fixed
points (foci F_{1}, F_{2}) is a

positive constant d(P, F_{1}) + d(P, F_{2}) = c

the center of the ellipse is the midpoint between the foci

the major axis is the line through the foci

the minor axis is the line
major axis which goes through the center point

the vertices are the points furthest from the center (where the major axis hits
the ellipse)

Note: an ellipse is symmetric with respect to its major axis, minor axis, and
center point

If the major axis of an ellipse is horizontal, its equation is:

its center is at (h, k), its foci are a distance c from
the center (in the x-direction) where c^2 = a^2 − b^2

the length of the major axis is 2a, the length of the minor axis is 2b

accordingly, the vertices are at (h + a, k) and (h − a, k)

If the major axis of an ellipse is vertical, its equation is:

its center is at (h, k), its foci are a distance c from
the center (in the y-direction) where c^2 = a^2 − b^2

the length of the major axis is 2a, the length of the minor axis is 2b

accordingly, the vertices are at (h, k + a) and (h, k − a)

again, you should be able to go between graphs, equations,
foci/vertices/major axis/minor axis/center

that is, given some little information you can produce the rest

10.3 Hyperbolas

a Hyperbola: the set of points (P), the difference of whose distances from two
fixed points (foci F_{1},

F_{2}) is a positive constant so d(P, F_{1}) − d(P, F_{2})
= c or d(P, F_{2}) − d(P, F_{1}) = c.

the center of the hyperbola is the midpoint between the foci

the transverse axis is the line through the foci

the conjugate axis is the line
transverse axis which goes through the center point

the vertices of the hyperbola are the points closest to the center (where the
transverse axis hits the

hyperbola)

Note: a hyperbola is symmetric with respect to its transverse axis, conjugate
axis and center

If the transverse axis of a hyperbola is horizontal, its equation is:

its center is at (h, k), its foci are a distance c from
the center (in the x-direction) where c^2 = a^2 + b^2

the length of the transverse axis is 2a, the length of the conjugate axis is 2b

accordingly, the vertices are at (h + a, k) and (h − a, k)

the equations for its asympototes are

If the transverse axis of a hyperbola is vertical, its equation is:

its center is at (h, k), its foci are a distance c from
the center (in the y-direction) where c^2 = a^2 + b^2

the length of the transverse axis is 2a, the length of the conjugate axis is 2b

accordingly, the vertices are at (h, k + a) and (h, k − a)

the equations for its asympototes are

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