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Advanced Algebra 2 Honors Final Review

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
  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

%%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

xf(x)
03
11
-11
  • 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
  1. 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
  6. \