Advanced Algebra 2 Honors Final Review
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
identity - every acceptable real number replacement for the variable is a solution
contradiction - no real number is a solution
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:
Analyze the problem
Pick a variable to represent the quantity being found
Find a way to express an amount in two ways
Form an equation
Solve
Answer
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:
The Trichotomy Property: for any real numbers a and b, one of the following statements is true: a<b, a=b, and a>b
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
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
ex. Graph f(x)=-2|x|+3
x | f(x) |
|---|---|
0 | 3 |
1 | 1 |
-1 | 1 |
graph these points
basic functions/parent functions: six common functions
f(x) = x
f(x) = x^2
f(x) = x^3
f(x) = √x
f(x) = cube root of x
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:
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
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:
Replace f(x) with y
Interchange the variables x and y
Solve the resulting equation for y
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
quadratic function: second-degree polynomial function
f(x)=ax^2+bx+c
To graph a quadratic function:
Determine if the parabola opens up or down
Find the vertex
Find the x-intercepts
Find the y-intercepts
Identify one additional point
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:
Find x and y intercepts
Determine end behavior
Make a sign chart and determine where the graph is above and below x-axis
Find symmetries
Plot points and draw graph as a smooth continuous curve
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
identity - every acceptable real number replacement for the variable is a solution
contradiction - no real number is a solution
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:
Analyze the problem
Pick a variable to represent the quantity being found
Find a way to express an amount in two ways
Form an equation
Solve
Answer
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:
The Trichotomy Property: for any real numbers a and b, one of the following statements is true: a<b, a=b, and a>b
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
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
ex. Graph f(x)=-2|x|+3
x | f(x) |
|---|---|
0 | 3 |
1 | 1 |
-1 | 1 |
graph these points
basic functions/parent functions: six common functions
f(x) = x
f(x) = x^2
f(x) = x^3
f(x) = √x
f(x) = cube root of x
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:
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
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:
Replace f(x) with y
Interchange the variables x and y
Solve the resulting equation for y
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
quadratic function: second-degree polynomial function
f(x)=ax^2+bx+c
To graph a quadratic function:
Determine if the parabola opens up or down
Find the vertex
Find the x-intercepts
Find the y-intercepts
Identify one additional point
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:
Find x and y intercepts
Determine end behavior
Make a sign chart and determine where the graph is above and below x-axis
Find symmetries
Plot points and draw graph as a smooth continuous curve
