Term
Looking at a graph, how can you tell if it is a function? 

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


Term
The output variable and the input variable:
Which one is dependent, and which is independent? 

Definition
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 


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

Definition
The domain is x (xaxis)
The range is f(x) or y (yaxis) 


Term
Pythagorean Theorem for a Right Triangle 

Definition
L^{2} + H^{2} = D^{2}
L = Length
H = Height
D = Diagonal 


Term

Definition


Term

Definition


Term
Vertical Shift of Function (up/down):
Horizontal Shift of Function (left/right): 

Definition
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(x5) will move right 5 units 


Term
Reflect Function Across xaxis:
Reflect Function Across yaxis: 

Definition
Reflect Function Across xaxis:
Multiply function by 1
EG: f(x) → f(x) will mirror across xaxis
Reflect Function Across yaxis:
Multiply x by 1
EG: f(x) → f(x) will mirror across yaxis 


Term
Vertically Stretch Graph of a Function:
Vertically Shrink Graph of a Function: 

Definition
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 


Term
Horizontally Stretch Graph of a Function:
Horizontally Shrink Graph of a Function: 

Definition
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 


Term

Definition
f(x) = f(x) is EVEN
(symmetry about the yaxis)
f(x) = f(x) not possible except for 0
(symmetry about the xaxis)
f(x) = f(x) and f(x) = f(x) are ODD
(symmetry about the origin) 


Term
How to find the inverse of a function: 

Definition
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) 


Term
Is this a function?
X: 3, 2, 4, 6, 8, 12
Y: 3, 3, 7, 12, 4, 8 

Definition
Yes  passes Vertical Line Test
All Domain values are unique 


Term
Is this a function?
X: 3, 2, 4, 3, 8, 12
Y: 3, 3, 7, 12, 4, 8 

Definition
No  does not pass Vertical Line Test
Domain contains duplicates (3 corresponds to two values in the range 3 and 12) 


Term
Slope of a Linear Function in terms of Rise and Run 

Definition
Rise
Slope = 
Run 


Term
Standard Form of a Linear Function 

Definition
y or f(x) = mx + b
m is the slope
b is the yintercept 


Term
How to calculate slope from coordinates of 2 points on the line: 

Definition
For (x_{1}, y_{1}) (x_{2}, y_{2})
y_{2}  y_{1}
M = 
x_{2}  x_{1} 


Term

Definition
y or f(x) = m(xx_{1}) + y_{1}
(xx_{1}) ends up being x
y_{1} ends up being b or yintercept 


Term
How to find the root of a linear function: 

Definition
Calculate y = mx + b as
0 = mx + b 


Term
Parallel lines have slopes that are ______
Perpendicular lines have slopes that are ______ 

Definition
Parallel lines have slopes that are EQUAL
EG: m_{1} = m_{2}
Perpendicular lines have slopes that are
NEGATIVELY RECIPROCAL
EG: m_{1} = 1/m_{2} or m_{2} = 1/m_{1} 


Term
How to find the point of intersection of 2 lines: 

Definition
For two lines y_{1}=m_{1}x_{1}+b_{1} and y_{2}=m_{2}x_{2}+b_{2}
b_{2}b_{1}
Point of intersection (x_{0}) is 
m_{1}m_{2}
(then can use this as x to find y) 


Term
In regression analysis,
r is ________
and r^{2} is ________ 

Definition
r is the CORRELATION COEFFICIENT
(a number between 1 and 1 that measures how well the best fitting line fits the data points)
r^{2} 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) 


Term
Standard form of a Quadratic Function 

Definition
y or f(x) = ax^{2}+bx+c
(a≠0, if a=0 then it is a horizontal line) 


Term
Standard Form vs. Vertex Form of a Quadratic Function 

Definition
Standard Form
y or f(x) = ax^{2}+bx+c
Vertex Form
y or f(x) = a(xh)^{2}+k
h,k are the x,y of the vertex 


Term
Move the vertex of
f(x) = 3x^{2}+1 to (3,2) 

Definition
Replace old x with new x
3x^{2} → 3(x+3)^{2}
remember, x moves the opposite way
Replace old y with new y
+1 → +2
3x^{2}+1 → 3(x+3)^{2} +2 


Term
Finding the Vertex of a Quadratic Function 

Definition
b
x = 
2a
Plug this into the equation to find y



Term
Finding the roots of a Quadratic Function: 

Definition
The root(s) are at
0 = ax^{2}+bx+c
Use the Quadratic Formula:
b ± √b^{2 } 4ac
x= 
2a 


Term
What is the Quadratic Formula?
What is it used for? 

Definition
Quadratic Formula
b ± √b^{2 } 4ac
x= 
2a
Quadratic Formula is used to find the roots of a quadratic function 


Term
What is the Discriminant and what can it tell you? 

Definition
The Discriminant is the b^{2 } 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 xaxis
If the Discriminant is negative, the graph does not intersect the xaxis (there is no root) 


Term
What kind of function is this:
f(x) = mx + b 

Definition


Term
Linear Regression Analysis
What is the correlation coefficient and how is it represented? 

Definition
correlation coefficient = r
Measures how well the best fitting line fits the data points. Ranges from 1 to 1. 


Term
Linear Regression Analysis
What is the coefficient of determination and how is it represented? 

Definition
Coefficient of Determination = r^{2} (the square of the correlation coefficient). Determines if the best fitting line can be used as a model (is it good enough?)
The closer r^{2} is to 1, the better the fit. 


Term
What kind of function is this:
f(x) = ax^{2} + bx + c 

Definition
Quadratic Function
(a ≠ 0)
The simplest form of a quadratic function is
f(x) = x^{2}
a.k.a. a power function 


Term
What kind of function is this:
ax^{4} + ax^{3} + ax^{2} + ax + a 

Definition
Polynomial Function
(of degree 4  quartic polynomial) 


Term
Standard form of a Polynomial Function 

Definition
ax^{4} + ax^{3} + ax^{2} + ax + a
(the exponent cannot be negative,
the exponent cannot be a fraction,
x cannot be in the denominator) 


Term
If the first (largest) term in a polynomial function is
ax^{4} the function is ____________
ax^{3} the function is ____________
ax^{2} the function is ____________
ax the function is ____________
ax^{0} ________________ 

Definition
If the first (largest) term in a polynomial function is
ax^{4} the function is quartic (parabola)
ax^{3} the function is cubic (snakelike)
ax^{2} the function is quadratic (parabola)
ax the function is linear (line)
ax^{0} is a horizontal line at y=a 


Term
Polynomial Function
bx^{4} + ax^{3} + ax^{2} + ax + g
What is b?
What is 4?
What is g?
What is bx^{4}?


Definition
b is the leading coefficient
4 is the degree/order
g is the constant term
bx^{4} is the leading term 


Term

Definition
f(x) = ax^{n}
is a monomial function
is a power function
(n > 0
b ≠ 0) 


Term
f(x) = ax^{n}
if n=0, graph is _________________
if n=1, graph is _________________
if n=2, graph is _________________
if n=3, graph is _________________ 

Definition
f(x) = ax^{n}
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) 


Term
Evenexponent Power Functions
x^{n} → n could equal _____
the shape is _______
graph gets ______ the _______ the exponent
When x>1 or x<1, ______ are ________
When 1>x>1, _______ are ________ 

Definition
x^{n} → 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 


Term
Oddexponent Power Functions
x^{n} → n could equal _____
the shape is _______
graph gets ______ the _______ the exponent
When x>1 or x<1, ______ are ________
When 1>x>1, _______ are ________ 

Definition
x^{n} → 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 


Term
Intermediate Value Theorem
(polynomial functions) 

Definition
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) 


Term
Factor Theorem
(polynomial functions) 

Definition
f(c) will equal zero ONLY IF (xc) is a factor of the polynomial.
In other words, the factors (xc) are the only places where the function will equal zero. 


Term

Definition
i^{2} = 1
√16 = √16 i = 4i 


Term
(xc)^{3} has a _________ of _____
if x=4, the factor of the polynomial is ______
if x = 3, the factor of the polynomial is ______ 

Definition
(xc)^{3} has a multiplicity of 3
if x=4, the factor of the polynomial is (x4)
if x = 3, the factor of the polynomial is (x+3) 


Term
(xc)^{3} will _____ the xaxis at the x=c
(xc)^{2} will _____ the xaxis at the x=c 

Definition
(xc)^{3} will cross the xaxis at the x=c
(xc)^{2} will touch the xaxis at the x=c 


Term
How do you represent a polynomial factor that does not cross or touch the xaxis anywhere? 

Definition
The constant factor k
f(x) = k(xc_{1})(xc_{2})(xc_{3})
Adding or subtracting from the constant factor k shifts the graph up or down the yaxis 


Term
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 

Definition
A polynomial of degree/order "n" can have a maximum of n roots
A polynomial of degree/order "n" can have a maximum of n1 turning points 


Term
Finding the rational (not irrational) zeros of a polynomial function:
Rational Zeros Theorem 

Definition
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 


Term
A quadratic function can have ____ turning points
A cubic function can have ____ turning points
A quartic function can have ____ turning points 

Definition
A quadratic function can have 1 turning point
A cubic function can have 2 turning points
A quartic function can have 3 turning points 


Term
Polynomial Functions
When the absolute value of x is large, end/longrun behavior of the graph will tend to ______ 

Definition
When the absolute value of x is large, end/longrun behavior of the graph will tend to follow the leading term 


Term
For every polynomial function where the degree is >0, there are complex numbers such that
f(x)=a(xc_{1}) (xc_{2})... etc...
(as long as a≠0)
This is known as _____________ 

Definition
The Linear Factorization Theorem
For every polynomial function where the degree is >0, there are complex numbers such that
f(x)=a(xc_{1}) (xc_{2})... etc...
(as long as a≠0) 


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

Definition
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. 


Term
What kind of function is this:
p(x)
f(x)= 
q(x) 

Definition


Term
What kind of function is this:
ax^{3}+bx^{2}+cx+d
f(x)= 
ax^{4}+bx^{3}+cx^{2}+dx+e 

Definition


Term
What is the domain of a rational function? 

Definition
The domain of a rational function is the set of all real numbers that are NOT roots of the denominator
(the denominator≠0) 


Term
(x2)(x+6)^{2}

(x2)(x6)
1. Root(s)/Zero(s):
2. Vertical Asymptote(s):
3. Hole(s):
4. Degree of numerator/denominator: 

Definition
(x2)(x+6)^{2}

(x2)(x6)
1. Root(s)/Zero(s): 6
2. Vertical Asymptote(s): 6
3. Hole(s): 2
4. Degree of numerator/denominator: 3/2 


Term
x^{3}+10x^{2}+12x72

x^{2}8x+12
1. Yintercept:
2. Horizontal Asymptote(s):
3. End behavior of graph:
4. Degree of numerator/denominator: 

Definition
x^{3}+10x^{2}+12x72

x^{2}8x+12
1. Yintercept: 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: x^{3}/x^{2} 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 


Term
Negative or positive? What degree?
[image] 

Definition


Term
Negative or positive? What degree?
[image] 

Definition


Term
Negative or positive? What degree?
[image] 

Definition


Term
Negative or positive? What degree?
[image]


Definition


Term
Asymptote of
a^{n}

b^{n} 

Definition
Horizontal asymptote at
y=a/b 


Term
Asymptote of
a^{n}

b^{N} 

Definition
Horizontal asymptote at
y=0 


Term
Asymptote of
a^{N}

b^{n} 

Definition
Oblique asymptote at
(divide the equation to find it) 


Term
Asymptote of
a^{NN}

b^{n} 

Definition

