Quarter 1 Test

Which point lies on the inverse of y+2=4(x-1) ?

(-2, 1)

Factor (3x)^3+1^3

(3x+1)(3x^2-3x+1) (3x+1)(9x^2-3x+1)

If the point (-7, 5) lies on the graph of y=f(x), what is a point on the inverse?

(5, -7)

Factoring sums and differences of cubes

(a same sign b)(a^2 opposite sign ab + b^2

Associative Property

(a+b)+c=a+(b+c) (a*b)c=a(b*c)

Write an equation with root 3i, where root 3i has a multiplicity of 2.

(x-3i)^2(x+3i)^2 AND THEN KEEP GOING BY MULTIPLYING THEM

Irrational and Complex roots with the quadratic formula

-Use quadratic formula when needed -Only use it when the highest degree is squared

Put these into ax^b form: 3/x^2 1/7x^3 8sqrt(x) 6/^3sqrt(x)

1. 3x^-2 2. 1/7x^-3 3. 8x^1/2 4.6x^-1/3

Steps to factor a number

1. Find all possible rational roots 2. Test all factors 3. Continue factoring or use synthetic division to factor completely and solve

How to multiply together imaginary roots such as (x+1-3i)(x+1+3i)

1. First, find the sum of them (1+3i + 1-3i). This gets you the x value. 2. Then, find the product of them (1+3i)(1-3i). This gets you the last value. x^2+2x+10

Factoring by grouping

1. Group the terms 2. Factor the GCF 3. Factor out the common binomial

The AC method for factoring trinomials

1. Multiply the leading coefficient times the constant term(last term). Remove the leading coefficient. 2. Factor this new trinomial. 3. Replace x with the original leading coefficient times x. 4. Factor 5. Divide by the original leading coefficient. (ALWAYS REDUCE EQUATION AS BEST YOU CAN BEFORE DOING THIS)

How to find an inverse equation STEPS.

1. Rewrite as y= 2. Switch the x and the y 3. solve for y 4. Change the y to f-1(x)=

State the domain of: y=x+3/x-9 y=sqrt(x-9) y=x^2-4/x+5

1. X=all real numbers/x=9 2. X > or equal to 9 3. X=all real numbers/x=5

Rules for factoring

1/x^2=x^-2 ^3sqrt(x^2)=x^2/3 ^2sqrt(x)=x^1/2

How many terms are in 2x^2+3x-7 ?

3 terms

Simplify 5(x)+9-(5(y)+9)/(x-y)

5x+9-5y-9/(x-y) =5x-5y/(x-y) =5(x-y)/(x-y) =5

What is a way to find possible solutions to sqrt(6-2x)

6-2x≥0 -2x≥-6 x≤3

Average rate of change from 0 to 4, if f(0)=1 and f(4)=7

7-1/4-0 =6/4 =3/2

3x^2= (xy^2)^2=

9x^2 x^2y^4

Expression

A combination of terms using addition or subtraction

To simplify you must have...

A common factor!

Consistent

A finite # of solutions

Piecewise function

A function defined by at least two equations, each of which applies to a different part of the domain.

One to one functions

A function in which each element of the range corresponds to one and only one element of the domain. So an element of the range can not have more than one element of the domain, and vice versa.

Even Functions

A function is even if it is... 1. symmetric over the y axis 2. f(-x)=f(x) (2 ARROWS UP OR DOWN)

Odd Functions

A function is odd if it is... 1. symmetric through the origin 2. f(-x)=-f(x) (1 ARROW UP 1 DOWN)

Existence of Inverse Functions

A function will have an inverse only if it is a one-to-one function. So a quick way to discover this is by using the horizontal line test and vertical line test.

Variable

A quantity represented by a letter that is unknown, unspecified, or can change within context of a problem.

Function

A relation in which each input value(x) is paired to only one output value(y). Each element of the domain is assigned to only one element of the range.

Linear Modeling

A scatter plot can be used to determine the strength of the relation or the correlation between data sets. The closer the data points fall along the line with a positive slope, the stronger the positive correlation and linear relationship.

Term

A single # or a combination of #s and variables using multiplication or division.

Lower Bounds

All roots are greater than that number. Finding: if you divide the dividend by the b in (x-b), and all the coefficients of the quotient alternate between positive and negative, it is a lower bound. Also, there can not be a remainder of 0.

Upper Bounds

All roots are less than that number. Finding: If you divide the dividend by the b in (x-b), and all numbers of the quotient are positive, it is an upper bound. Also, there can not be a remainder of 0.

What should you remember when finding complicated compositions?

Always plug the represented "x" value, whether it be x, 3x+10 or 300, into your equation value. For ex: if it is f(x), plug x into the x of f's equation. If it is f(3), plug 3 into the x of f's equation.

Polynomial Function

Anything with exponential power greater than 0. The graphs of them are continuous and smooth.

Descartes Chart

Degree # positive real roots(can be multiple) #negative real roots(can be multiple) complex/imaginary roots zero roots

Long division answer

Dividend=divisor*quotient + remainder (dividend is what you are dividing, quotient is the answer, divisor is what you are dividing it by, and remainder is what is left over.

Synthetic Division

Divisor must be in x-a form (x-5 means a is 5, x+2 means a is -2)

Linear Equations

Equations where the highest power of the variable is 1, or highest degree of x is 1. This is also called a first degree equation.

The number of complex roots always must be...

Even. If they are not, one of them is a 0 root.

Simplifying Radicals

Find the perfect squares that go into it, and simplify. ex: squareroot(80) squareroot(16*5) 4squareroot(5)

Rational Root Theorem

For any polynomial, the possible factors are... positive and negative factors of the constant/positive and negative factors of the leading coefficient

What does a closed circle mean?

Greater than and equal to or less than and equal to

What does a open circle mean?

Greater than or less than

Vertical Line Test

If a vertical line drawn on a graph intersects the graph more than once, it is NOT a function.

What makes a number undefined?

If the number has a zero in the denominator, or if it has a negative in the radical.

Factor Theorem

If you plug "a" into the equation and get 0 as a remainder, then (x-a) is a factor of the equation.

Multiplicity

In general: For a factor (x-c)^m If m is even-hits but does not cross x axis if m is odd-crosses x axis

Composition of functions

In general: (k(f(g(4))) So first you do g(4), then you do f(g(4)), then k(f(g(4))).

What are inverses of a line a reflection over?

Inverses are a reflection over the y=x line.

Conjugate Pairs

Irrational and Complex roots always appear in conjugate pairs ex: 3+i and 3-i 4 + sqrt(5) and 4-sqrt(5)

Slope is a characteristic of ____ functions only

LINEAR

How does the coefficient affect the graph?

Larger values of the coefficient squash it towards the y axis, smaller values expand it away from the y axis.

Inconsistent

No solutions

How to find # turning points?

One less than the degree!

ALWAYS REMEMBER YOUR

Parenthesis when plugging something an expression in for x, or just in general. For example 8-4x+8 is MUCH different from 8-(4x+8). In one you distribute, in the other you don't.

Finding intersection points

Set the equations equal to each other

Steps to solving systems of linear equations

Step 1: Simplify and put all equations into the Ax+By+Cz=D format. Step 2: Choose to eliminate any one of the variables from any pair of two equations. Step 3: Eliminate the same variable chosen from step 2 from any other pair f equations, creating a system of two equations and two unknowns. Step 4: Solve the remaining system found in step 2 and 3. Step 5: Solve for the second variable by plugging into one of the equations found in step 2 or 3. Step 6: Solve for the third variable by plugging into one of the original equations. Step 7: Check

What happens to points in inverses?(for example what are the points on the inverse line of (-3,5))

The domain and range swap in the inverse. For example, the points in the inverse would be (5,-3).

Absolute min and max

The highest maximum and lowest minimum in the function.

How do you find the # of real negative roots? (descartes)

The number of real negative roots is equal to the number of sign changes in f(-x) or multiples of 2 less than that number. Remember for finding f(-x), if the exponent is odd the sign switches and if it is even it does not.

How do you find the # of real positive roots? (descartes)

The number of real positive roots is equal to the number of sign changes, and then the multiples of two less than that number.

Domain

The set of all the input(x) values in the relation.

Range

The set of all the output(y) values in the relation.

If f(g(x))-x and g(f(x))=x, then what are g and f?

They are inverses. Inverses tend to "undo" each other. So if f and g are inverses, the f(g(3))=3 and f(g(10x)=10x.

Relative Min and Max

They are minimums and maximums that must have points surrounding them on either side.

Identity

True for all values of x (all real numbers)

Like terms

Two or more terms that have the same variables raised to the same powers. In like terms, only the coefficients can differ.

The Remainder Theorem

When you plug "a" into the original equation, you will get the remainder. This is a good way to check your work.

Roots of a polynomial equation?

X intercepts or zeros

does x^2-x count as a sign change?

YES

When finding the domain of f(g(x))

You must consider the domains of BOTH g(x) and of f(g(x)). So first find the domain of g(x), then solve for f(g(x) and consider both!

How does slope apply to non linear functions?

You use average rate of change!

If f(a)=-b, then f-1(-b)=

a

When finding the domain preserve...

both equations into your answer. For example, if k(x)=sqrt(8-x), f(x)=4x+8 and you are solving to find the domain of k(f(x), you can leave the equation as sqrt(8-(4x+8)). So your domain would be x≤0.

Distributive Property

c(a+b)=c*a+c*b

WHEN SETTING UP SYNTHETIC DIVISION REMEMBER TO

check for any missing exponents (jump from x^3 to x) and use zeroes to fill them in.

If f(x)=x^2, find the average rate of change from 0 to 2.

f(2)-f(0)/2-0 =4-0/2 =2

What is f(x) and what is x?

f(x) is y. So find when f(x)=3, means find all of the x values for when y=3. x is x. So find f(3), means find the y value for when x=3.

If f(x)=3x+5, and g(x)=x^2+1, find g(f(x)).

g(f(x)). Step 1. Plug in the "x" into your equation (f) f(x)=3x+5 g(3x+5) Step 2. Plug your new "x" (3x+5), into your equation (g). g(3x+5)=(3x+5)^2+1 Step 3. Solve f(g(x))=9x^2+30x+25

Although the y values can have more than one x value, the x values can not

have more than one y value!

Degree of a polynomial?

highest exponent

End Behavior

n is the exponent, and an is the leading coefficient.

What is interval notation?

the one with the brackets and perenthesis.

How would you simplify x(x^2-1)/(x-1)

x(x+1)(x-1)/(x-1)=x(x+1)

What is a way to find the possible solutions to x-2/x+3

x+3≠0 x≠-3 x=R/x=-3

Commutative Property

x+y=y+x xy=yx

Laura is thinking of a number such that the sum of the number and five times two more than the number is 26 more than four times the number. Determine the number.

x=8

Long division example: x^3+6x^2-4x-24/x+2

x^3+6x^2-4x-24=(x+2)(x^2+4x-12x)+0

Point-Slope formula

y-y1=m(x-x1) (use when we know the slope and a point on the line)

Slope-Intercept formula

y=mx+b (use when we know the slope and y intercept of a line)