Chapter 1

equation: a statement indicating that two quantities are equal

the solution set: the set of all solutions of an equation

• there can be restrictions on the values of a variable

• ex. x^2+4/x-2. x cannot be 2 because then the denominator would be 0

Types of equations

1. identity - every acceptable real number replacement for the variable is a solution

2. contradiction - no real number is a solution

3. solution set contains some but not all real numbers

Linear equations: ax+b = 0

Rational equations: equations that contain rational expressions

mathematical modeling: the process of finding the equations that describe the words of the problem

Steps:

1. Analyze the problem

2. Pick a variable to represent the quantity being found

3. Find a way to express an amount in two ways

4. Form an equation

5. Solve

6. Answer

7. Check

complex numbers: numbers that are the sum or difference of a real number and an imaginary number

imaginary numbers: denoted by i

• i = square root of -1

complex conjugates: complex numbers with different signs

• a+bi and a- bi

The Powers of I:

i = √-1

i^2 = -1

i^3 = -i

i^4 = 1

• the pattern is i, -1, -i, 1

If n is a natural number that has a remainder of r when divided by 4, i^n = i^r

quadratic equation: an equation that can be written in the form ax^2+bx+c=0 where a, b, and c are real numbers and a doesn’t = 0

Zero Factor Theorem: If a and b are real numbers and if ab = 0, then a=0 or b=0

Square Root Property: If x^2 =c, the x=√c 0r x=-√c

Quadratic Formula: x = -b + √b^2-4ac / 2a

discriminant: the value b^2 -4ac

Polynomial Equation: arrange terms of polynomial in descending order based on the degree and set equal to zero

• a>b if point a lies to the right of point b on a number line

Properties of Inequalities:

1. The Trichotomy Property: for any real numbers a and b, one of the following statements is true: a<b, a=b, and a>b

2. The Transitive Property: If a, b, and c are real numbers, then: if a<b, then a+c<b+c, a-c<b-c

equivalent inequalities: inequalities with the same solution set

compound inequalities: two inequalities put together

• Solve 5<3x-7<=8

Add 7 to all sides

12<3x<=15 divided by 3

4<x<=5 (4,5 ]

quadratic inequalities: if a doesn’t equal 0, inequalities like ax^2+bx+c<0 and ax^2+bx+c>=0

rational inequalities: inequalities that contain fractions with polynomial numerators and denominators

absolute value: |x|

|-7| - 7

Chapter 2

relation: a correspondence between a set of input values and a set of output values

domain: input values of x

range: output values of y

function: any relation that assigns exactly one output value for each input value

• y is dependent and x is independent

• represented through verbal expression, table, ordered pairs, equation, or a graph

function notation: y= f(x)

difference quotient: f(x+h)-f(x)/ h

rectangular coordinate system: 2 perpendicular number lines that divide the plane into four quadrants

Linear Equations: Ax+By=C and y = mx+b

• to find y-intercept, replace x with 0.

• horizontal and vertical lines are not functions

linear function: a function in standard form ax+b=f(x)

Distance Formula

d= √(x2-x1)^2 + (y2-y1)^2

ex. Find the distance between (-1,-2) and (-7,8)

√(-7--1)^2 + (8--2)^

√-6^2+10^2

√136

= 2√34

Midpoint Formula

(x1+x2/2 , y1+y2/2)

ex. Find the midpoint of (-7,2) and (1,-4)

(-7+1/2 , 2+-4/2)

(-6/2, -2/2)

(-3,-1)

slope: the measure of the steepness of a line

• constant rate of change

• change in y over the change in x

• the slope of a horizontal line is 0

• the slope of a vertical line is undefined

slope intercept form: y=mx+b

point slope form: y-y1=m(x-x1)

Parabolas: lowest point is the vertex, symmetric about the y-axis

How to test symmetry:

• to test the x-axis, replace y with -y

• to test the y-axis, replace x with -x

• to test origin, replace y with -y and x with -x

circle: the set of all points in a plane that are a fixed distance from the center

radius: fixed distance from the center to a point on a circle

equation of a circle: (x-h)^2+(y-k)^2=r^2

general equation: ax^2+by^2+cx+dy+e=0

ratio: the quotient of two numbers

proportion: an equation indicating that two ratios are equal

• 2/3 = 4/6

• in the proportion a/b = c/d, a and d are the extremes, b and c are the means

• ad=bc

ex. solve the proportion x/5 = 2/x+3

x(x+3) = 2(5)

x^2+3x=10

x^2+3x-10=0

(x+5)(x-2)=0

x=-5,2

hyperbola: the graph of the equation y=k/x

direct variation: when a variable’s ratio is a constant, y/x=k or y=kx

inversely variation: when a variable’s product is a constant, xy=k

joint variation: when y varies jointly with x, y=kwx

Chapter 3

ex. Graph f(x)=-2|x|+3

• graph these points

basic functions/parent functions: six common functions

1. f(x) = x

2. f(x) = x^2

3. f(x) = x^3

4. f(x) = √x

5. f(x) = cube root of x

6. f(x) = |x|

• the graph of y=f(x)+k is the same as y=f(x), but translated k units up

• the graph of y=f(x)-k is the same as y=f(x), but translated k units down

even function: symmetric about the y-axis - if f(-x) is f(x)

odd function: symmetric about the origin

local maximum: y value of a high point

local minima: y value of a low point

• extrema

• local max. occurs where the function changes from increasing to decreasing

• local min. occurs where the function changes from decreasing to increasing

If the ranges of functions f and g are subsets of real #s, then

• (f+g)(x)=f(x)g(x)

• (f*g)(x)=f(x)g(x)

• (f/g)(x)=f(x)g(x)

• (f-g)(x)=f(x)g(x)

composition functions: when one quantity is a function of a second quantity that depends on a third quantity

ex. f(x)=2x+7 g(x)=x^2-1. Find (f*g)(x)

2(x^2-1)+7

2x^2-2+7

2x^2+5

decomposition: reversing the composition process and using a function h

one-to-one functions: functions whose inverses are also functions

ex. Determine if the function is one-to-one: f(x)=x^4+x^2

No, because different numbers can have the same outputs - 2 and -2 have the same output

horizontal line test: used to determine if the graph is one-to-one

Properties of One-to-One functions:

1. If f is a one-to-one function, there is a one-to-one function f^-1(x) such that

• (f^-1◦f)(x)=x and (f◦f^-1)(x)=x

2. The domain of f is the range of f^-1 and the range of f is the domain of f^-1

Finding the Inverse Functions:

1. Replace f(x) with y

2. Interchange the variables x and y

3. Solve the resulting equation for y

4. Replace y with f^-1(x)

ex. Find the inverse of f(x)=3/2x+2

2(x)=3/2y+2(2)

2x=3y+4

2x-4=3y

y=2x-4/3

f^-1(x)=2x-4/3

• the graph of a function and its inverse are reflections of each other about the line y=x

Chapter 4

quadratic function: second-degree polynomial function

• f(x)=ax^2+bx+c

To graph a quadratic function:

1. Determine if the parabola opens up or down

2. Find the vertex

3. Find the x-intercepts

4. Find the y-intercepts

5. Identify one additional point

6. Draw a curved line through these points

Polynomial Functions:

• cannot have a fraction, radical, negative exponent, or an absolute value

• the graph is a smooth continuous curve

zero: a value of x for which f(x)=0

power function: polynomial function in the form f(x)=a^n

• end behavior is the same as the graph of its term with the highest degree

To graph of polynomial functions:

1. Find x and y intercepts

2. Determine end behavior

3. Make a sign chart and determine where the graph is above and below x-axis

4. Find symmetries

5. Plot points and draw graph as a smooth continuous curve