Vertical Line Test:

Vertical line should not intersect graph in more than one place. If it does, then it is not a function.

The output variable and the input variable:

Which one is dependent, and which is independent?

The output variable is dependent

(it depends upon the input)

The input variable is independent

(may have its value freely chosen regardless of any other variable values)

The output is a function of (depends upon) the input

The DOMAIN (or INPUT) is on the _____ axis.

The RANGE (or OUTPUT) is on the _____ axis.

The domain is x (x-axis)

The range is f(x) or y (y-axis)

L² + H² = D²

L = Length

H = Height

D = Diagonal

(B x H)

---------

2

Vertical Shift of Function (up/down):

Horizontal Shift of Function (left/right):

Vertical Shift of Function (up/down):

Add or subtract from the function

EG: f(x) → f(x) + 5 will move up 5 units

Horizontal Shift of Function (left/right):

Add or subtract the reverse from x

EG: f(x) → f(x-5) will move right 5 units

Reflect Function Across x-axis:

Reflect Function Across y-axis:

Reflect Function Across x-axis:

Multiply function by -1

EG: f(x) → -f(x) will mirror across x-axis

Reflect Function Across y-axis:

Multiply x by -1

EG: f(x) → f(-x) will mirror across y-axis

Vertically Stretch Graph of a Function:

Vertically Shrink Graph of a Function:

Vertically Stretch Graph of a Function:

Multiply function by a number greater than 1

EG: f(x) → 3f(x) will vertically stretch the graph

Vertically Shrink Graph of a Function:

Multiply function by a number between 0 and 1

EG: f(x) → 0.5f(x) will vertically shrink the graph

Horizontally Stretch Graph of a Function:

Horizontally Shrink Graph of a Function:

Horizontally Stretch Graph of a Function:

Multiply x by a number between 0 and 1

EG: f(x) → f(0.5x) will horizontally stretch the graph

Horizontally Shrink Graph of a Function:

Multiply x by a number greater than 1

EG: f(x) → f(3x) will horizontally shrink the graph

f(x) = f(-x) is EVEN

(symmetry about the y-axis)

f(x) = -f(x) not possible except for 0

(symmetry about the x-axis)

-f(x) = f(-x) and f(-x) = -f(x) are ODD

(symmetry about the origin)

1. Replace f(x) with y

2. Solve for x in terms of y (x on one side, alone)

3. Interchange x and y, then replace y with f^-1(x)

Is this a function?

X: 3, 2, 4, 6, 8, 12

Y: 3, 3, 7, 12, 4, 8

Yes - passes Vertical Line Test

All Domain values are unique

Is this a function?

X: 3, 2, 4, 3, 8, 12

Y: 3, 3, 7, 12, 4, 8

No - does not pass Vertical Line Test

Domain contains duplicates (3 corresponds to two values in the range- 3 and 12)

          Rise

Slope = ----------

          Run

y or f(x) = mx + b

m is the slope

b is the y-intercept

For (x₁, y₁) (x₂, y₂)

     y₂ - y₁

M = -----------

     x₂ - x₁

y or f(x) = m(x-x₁) + y₁

(x-x₁) ends up being x

y₁ ends up being b or y-intercept

Calculate y = mx + b as

0 = mx + b

Parallel lines have slopes that are ______

Perpendicular lines have slopes that are ______

Parallel lines have slopes that are EQUAL

EG: m₁ = m₂

Perpendicular lines have slopes that are

NEGATIVELY RECIPROCAL

EG: m₁ = -1/m₂   or   m₂ = -1/m₁

For two lines     y₁=m₁x₁+b₁    and    y₂=m₂x₂+b₂

                               b₂-b₁

Point of intersection (x₀) is ------------

                               m₁-m₂

(then can use this as x to find y)

In regression analysis,

r is ________

and r² is ________

r is the CORRELATION COEFFICIENT

(a number between -1 and 1 that measures how well the best fitting line fits the data points)

r² is the COEFFICIENT OF DETERMINATION

(a number that determines if the best fitting line can be used as a data model. Closer to 1, the better the fit)

y or f(x) = ax²+bx+c

(a≠0, if a=0 then it is a horizontal line)

Standard Form

y or f(x) = ax²+bx+c

Vertex Form

y or f(x) = a(x-h)²+k

-h,k are the x,y of the vertex

Move the vertex of

f(x) = 3x²+1 to (-3,2)

Replace old x with new x

3x² → 3(x+3)²

remember, x moves the opposite way

Replace old y with new y

+1 → +2

3x²+1 → 3(x+3)² +2

    -b

x = ------

     2a

Plug this into the equation to find y

The root(s) are at

0 = ax²+bx+c

Use the Quadratic Formula:

    -b ± √b² - 4ac

x= --------------------

    2a

What is the Quadratic Formula?

What is it used for?

Quadratic Formula

    -b ± √b² - 4ac

x= --------------------

    2a

Quadratic Formula is used to find the roots of a quadratic function

The Discriminant is the b² - 4ac part of the Quadratic Function

If the Discriminant is positive, there are two roots

If the Discriminant is zero, there is one root, the graph is sitting on the x-axis

If the Discriminant is negative, the graph does not intersect the x-axis (there is no root)

What kind of function is this:

f(x) = mx + b

Linear Regression Analysis

What is the correlation coefficient and how is it represented?

correlation coefficient = r

Measures how well the best fitting line fits the data points. Ranges from -1 to 1.

Linear Regression Analysis

What is the coefficient of determination and how is it represented?

Coefficient of Determination = r² (the square of the correlation coefficient). Determines if the best fitting line can be used as a model (is it good enough?)

The closer r² is to 1, the better the fit.

What kind of function is this:

f(x) = ax² + bx + c

Quadratic Function

(a ≠ 0)

The simplest form of a quadratic function is

f(x) = x²

a.k.a. a power function

What kind of function is this:

ax⁴ + ax³ + ax² + ax + a

Polynomial Function

(of degree 4 - quartic polynomial)

ax⁴ + ax³ + ax² + ax + a

(the exponent cannot be negative,

the exponent cannot be a fraction,

x cannot be in the denominator)

If the first (largest) term in a polynomial function is

ax⁴ the function is ____________

ax³ the function is ____________

ax² the function is ____________

ax the function is  ____________

ax⁰ ________________

If the first (largest) term in a polynomial function is

ax⁴ the function is quartic (parabola)

ax³ the function is cubic (snakelike)

ax² the function is quadratic (parabola)

ax the function is linear (line)

ax⁰ is a horizontal line at y=a

Polynomial Function

bx⁴ + ax³ + ax² + ax + g

What is b?

What is 4?

What is g?

What is bx⁴?

b is the leading coefficient

4 is the degree/order

g is the constant term

bx⁴ is the leading term

f(x) = axⁿ

is a monomial function

is a power function

(n > 0

b ≠ 0)

f(x) = axⁿ

if n=0, graph is _________________

if n=1, graph is _________________

if n=2, graph is _________________

if n=3, graph is _________________

f(x) = axⁿ

if n=0, graph is a horizontal line at y=a

if n=1, graph is linear with slope of a (odd function)

if n=2, graph is parabola, branches facing up when a is a is positive, down when a is negative (even function)

if n=3, graph is snakelike, increasing when a is positive, decreasing when a is negative (odd function)

Even-exponent Power Functions

xⁿ → n could equal _____

the shape is _______

graph gets ______ the _______ the exponent

When x>1 or x<-1, ______ are ________

When -1>x>1, _______ are ________

xⁿ → n could equal 2, 4, etc.

the shape is a parabola

graph gets flatter (on the bottom) the higher the exponent

When x>1 or x<-1, branches are steeper

When -1>x>1, branches are flatter

Odd-exponent Power Functions

xⁿ → n could equal _____

the shape is _______

graph gets ______ the _______ the exponent

When x>1 or x<-1, ______ are ________

When -1>x>1, _______ are ________

xⁿ → n could equal 1, 3, 5, etc.

the shape is snakelike

graph gets flatter (on the bottom) the higher the exponent

When x>1 or x<-1, traces are steeper

When -1>x>1, traces are flatter

Intermediate Value Theorem

(polynomial functions)

If the result of f(a) and f(b) are opposite signs (+/-), then there must be at least one root between them

(as long as a≠b)

Factor Theorem

(polynomial functions)

f(c) will equal zero ONLY IF (x-c) is a factor of the polynomial.

In other words, the factors (x-c) are the only places where the function will equal zero.

i² =

√-16 =

i² = -1

√-16 = √16 i = 4i

(x-c)³ has a _________ of _____

if x=4, the factor of the polynomial is ______

if x = -3, the factor of the polynomial is ______

(x-c)³ has a multiplicity of 3

if x=4, the factor of the polynomial is (x-4)

if x = -3, the factor of the polynomial is (x+3)

(x-c)³ will _____ the x-axis at the x=c

(x-c)² will _____ the x-axis at the x=c

(x-c)³ will cross the x-axis at the x=c

(x-c)² will touch the x-axis at the x=c

The constant factor k

f(x) = k(x-c₁)(x-c₂)(x-c₃)

Adding or subtracting from the constant factor k shifts the graph up or down the y-axis

A polynomial of degree/order "n" can have a maximum of ___ roots

A polynomial of degree/order "n" can have a maximum of ___ turning points

A polynomial of degree/order "n" can have a maximum of n roots

A polynomial of degree/order "n" can have a maximum of n-1 turning points

Finding the rational (not irrational) zeros of a polynomial function:

Rational Zeros Theorem

p (all rational factors of constant term)     

___                                                         

r (all rational factors of leading coefficient)

any of these that lead to f(x)=0 are the rational zeros

A quadratic function can have ____ turning points

A cubic function can have ____ turning points

A quartic function can have ____ turning points

A quadratic function can have 1 turning point

A cubic function can have 2 turning points

A quartic function can have 3 turning points

Polynomial Functions

When the absolute value of x is large, end/long-run behavior of the graph will tend to ______

For every polynomial function where the degree is >0, there are complex numbers such that

f(x)=a(x-c₁) (x-c₂)... etc...

(as long as a≠0)

This is known as _____________

The Linear Factorization Theorem

For every polynomial function where the degree is >0, there are complex numbers such that

f(x)=a(x-c₁) (x-c₂)... etc...

(as long as a≠0)

Every polynomial of a degree of ≥1 with complex coefficients has at least one zero in the complex number system.

This is called _______________

The Fundamental Theorem of Algebra

Every polynomial of a degree of ≥1 with complex coefficients has at least one zero in the complex number system.

What kind of function is this:

        p(x)

f(x)= -----------

        q(x)

What kind of function is this:

        ax³+bx²+cx+d

f(x)= -------------------------

        ax⁴+bx³+cx²+dx+e

The domain of a rational function is the set of all real numbers that are NOT roots of the denominator

(the denominator≠0)

(x-2)(x+6)²

----------------

(x-2)(x-6)

1. Root(s)/Zero(s):

2. Vertical Asymptote(s):

3. Hole(s):

4. Degree of numerator/denominator:

(x-2)(x+6)²

----------------

(x-2)(x-6)

1. Root(s)/Zero(s): -6

2. Vertical Asymptote(s): 6

3. Hole(s): 2

4. Degree of numerator/denominator: 3/2

x³+10x²+12x-72

-----------------------

x²-8x+12

1. Y-intercept:

2. Horizontal Asymptote(s):

3. End behavior of graph:

4. Degree of numerator/denominator:

x³+10x²+12x-72

-----------------------

x²-8x+12

1. Y-intercept: x=0 is not a root of the denominator, so evaluate function at x=0. y=-6

2. Horizontal Asymptote(s): oblique asymptote, divide the equation to find it. x+18

3. End behavior of graph: x³/x² which would be a line increasing as x increasing that crosses the graph at x=2 and x=6

4. Degree of numerator/denominator: 3/2

Negative or positive? What degree?

[image]

Negative

Odd

Negative or positive? What degree?

[image]

Positive

Odd

Negative or positive? What degree?

[image]

Negative

Even

Negative or positive? What degree?

 [image]

Positive

Even

Asymptote of

aⁿ

-------

bⁿ

Horizontal asymptote at

y=a/b

Asymptote of

aⁿ

------

b^N

Horizontal asymptote at

y=0

Asymptote of

a^N

------

bⁿ

Oblique asymptote at

(divide the equation to find it)

Asymptote of

a^NN

---------

bⁿ