all x's must be different, x's are used one time only,use vertical line test on the graph
1/(2^3) = 1/8
y=kxz, y varies jointly with x and z
horizontal asymptotes for rational expressions
(BOSTON)-if degree of Bottom is bigger, y=0; if degrees are same for numerator and denominator, use coefficients a/c; if Top has a larger degree, NO horizontal asymptote
vertical asymptoptes for rational expressions
set denominator equal to zero and factor; remember to divide out any common factors from numerator!
x-intercepts for rational expressions
set numerator equal to zero and factor, remember to divide out any common factors from denominator!
How do you factor a sum or difference of cubes?
first one, second one, first one squared, product of the two and the last one squared, first sign's the same, second one's not, last one's always plus and here's what you've got!
A = Pe^(rt)
What's the square root of -1?
What does the rational exponent mean?
power/root = "power over root"
How do you find the inverse of a function?
1. "y = " form, 2. switch x and y, 3. solve for y
How do you solve an absolute value equation?
Set the expression from inside the absolute value equal to the positive constant and to the negative constant.
What ways can you solve a system?
1. graph, 2. substitution, 3. linear combination or elimination, 4. use a matrix equation, if the system is linear
Solving Absolute Value Inequalities
GOLA; Greater than - OR, Less than - AND
Compound Interest Formula
A = P(1 + r/n)^(n x t), r is the rate, n is the number of times compounded, t is time
Standard Form for a Quadratic
f(x) = ax^2 + bx + c
Vertex Form for a Quadratic
y = a(x - h)^2 + k
Intercept Form for a Quadratic
y = a(x - p)(x - q)
Standard Form for a Line
Ax + By = C
Slope-Intercept Form for a Line
y = mx + b
m = (y (sub 2) - y (sub 1))/(x (sub 2) - x(sub 1)), difference of the y's over the difference of the x's
y - y(sub1) = m(x - x(sub 1))
x = (-b plus or minus) the square root of b^2 -4ac) / 2a from y = ax^2 + bx + c
Completing the Square
Remember to have a coefficient of 1 for the squared term. Take 1/2 of b and square it. Add to both sides. Solve through square roots.
Methods to Solve a Quadratic
1. Graphing, 2. Factoring, if factorable, 3. Completing the Square, 4. Quadratic Formula
Standard Form Equation for a Circle with center (h, k)
(x - h)^2 + (y - k)^2 = r^2, circle with center at (h, k) and radius of r
Standard Form Equation for a Circle with center at the origin
x^2 + y^2 = r^2
Domain for an Exponential Function?
All real numbers
Range for a Logarithmic Function?
All real numbers
Rectangular array of numbers in rows and columns
What must be true to add two matrices?
The dimensions must be the same. (Rows and Columns)
What must be true to multiply two matrices?
The columns of the first must match the rows of the second matrix.
When solving a matrix equation, use...?
"A inverse times B" meaning A^-1 times B
Does every matrix have an inverse?
No. If the determinant equals 0, there will not be an inverse.
What type of matrix must you have to take a determinant?
A square matrix
Consists of one row
Consists of one column
What do you know about slopes of parallel lines?
Parallel lines have the same slope!
What do you know about slopes of perpendicular lines?
Perpendicular lines have slopes that are opposite reciprocals of each other!
What is the slope for a horizontal line?
What is the discriminant?
b^2 minus 4ac
What does the value of the discriminant tell about a quadratic?
If the discriminant = 0, there is one real solution & graph "sits or bumps" x-axis. If the discriminant > than 0, there are two real solutions & graph crosses x-axis twice. If the discriminant < 0, there are no real solutions, two imaginary solutions & graph does not cross x-axis.
Write log base a of b = c in exponential form .
a ^ c = b
log base a of x + log base a of y =
log base a of (x times y) This is the product property of logarithms.
log base a of x - log base a of y =
log base a of (x divided by y) This is the quotient property of logarithms.
x log base n of y =
log base n of y ^ x This is the power property of logarithms.
Area of a triangle given coordinates for the three vertices?
Enter three vertices, (x, y) in a 3 X 3 matrix, in rows 1, 2, and 3, with 1's in the last column. Take plus or minus 1/2 of the determinant.
How do you set up a matrix equation to solve a linear system of equations?
There are three matrices in the equation. A coefficient matrix, where coefficients come from standard form equations, a variable matrix, and a constant matrix.
First degree polynomials are called...
Second degree polynomials are called...
Third degree polynomials are called...
Fourth degree polynomials are called...
Absolute Value Equation with vertex (h, k)
y = a(x - h)^2 + k
Composition of Functions
f ( g(x)) means that function g is the input to function f
How many roots does a square root have?
Two, positive and negative roots
Can you take a cube root of a negative number?
Yes, and the answer will be negative. A negative number raised to an odd power equals a negative.
What's the horizontal line test?
Used on an original function to determine if the inverse would be a function. Yes, you may still use the vertical line test on an inverse.
How do you know that two functions are inverses of each other?
f ( g(x)) = g (f(x)) = x
What is the line of reflection for inverses?
y = x
What is a determinant?
A real number associated with a square matrix.
What is a circle?
The set of all points (x, y) that are equidistant from a fixed point called the center.