%%equation:%% a statement indicating that two quantities are equal
%%the solution set%%: the set of all solutions of an equation
Types of equations
%%Linear equations:%% ax+b = 0
%%Rational equations:%% equations that contain rational expressions
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%%mathematical modeling:%% the process of finding the equations that describe the words of the problem
Steps:
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%%complex numbers:%% numbers that are the sum or difference of a real number and an imaginary number
%%imaginary numbers:%% denoted by i
%%complex conjugates%%: complex numbers with different signs
The Powers of I:
i = â-1
i^2 = -1
i^3 = -i
i^4 = 1
If n is a natural number that has a remainder of r when divided by 4, i^n = i^r
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%%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 doesnât = 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=âc 0r x=-âc
Quadratic Formula: x = -b + âb^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
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Properties of Inequalities:
%%equivalent inequalities%%: inequalities with the same solution set
%%compound inequalities:%% two inequalities put together
Add 7 to all sides
12<3x<=15 divided by 3
4<x<=5 (4,5 ]
%%quadratic inequalities:%% if a doesnât equal 0, inequalities like ax^2+bx+c
%%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
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
%%linear function%%: a function in standard form ax+b=f(x)
Distance Formula
%%d= â(x2-x1)^2 + (y2-y1)^2%%
ex. Find the distance between (-1,-2) and (-7,8)
â(-7--1)^2 + (8--2)^
â-6^2+10^2
â136
= 2â34
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
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
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How to test symmetry:
%%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
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%%ratio:%% the quotient of two numbers
%%proportion:%% an equation indicating that two ratios are equal
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 variableâs ratio is a constant, y/x=k or y=kx
%%inversely variation:%% when a variableâs 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 |
%%basic functions/parent functions:%% six common functions
%%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
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If the ranges of functions f and g are subsets of real #s, then
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
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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:
Finding the Inverse Functions:
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
quadratic function: second-degree polynomial function
To graph a quadratic function:
Polynomial Functions:
zero: a value of x for which f(x)=0
power function: polynomial function in the form f(x)=a^n
To graph of polynomial functions:
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