Rectangles
Corners equal 360 degrees.
Area = Base x Height
Area = Base x Height
Triangles
Corners equal 180 degrees
Area equals 1/2 Base x Height
Area equals 1/2 Base x Height
Circles
360 degrees
Circumference = 2πr (r = radius)
Area = πr^2 (squared)
Circumference = 2πr (r = radius)
Area = πr^2 (squared)
π (Pi)
π is the ratio of the circumference to the radius of a circle. (circumference/radius) π is a mathematical constant.
Pythagorean Theorem
For a right triangle, A^2 + B^2 = C^2, where C is the hypotenuse and A and B are the other two legs of the right triangle.
Order of Operations
Parentheses
Exponents (and roots)
Multiplications
Division
Addition
Subtraction
Exponents (and roots)
Multiplications
Division
Addition
Subtraction
X-axis vs. Y-axis
Remember "X to the Left, Y to the sky!" X is horizontal and Y is the vertical.
Distance Formula
Midpoint Formula
Slope
Slope = Rise/Run Slope is often represented as "m"
Standard form of a Linear Equation
ax + by = c
Slope-Intercept Form of a Linear Equation
y = mx + b (m = the slope and b = the y-intercept.)
Point Slope Form of a Linear Equation
Intercept Form
x/a + y/b = 1 (a is the x-intercept and b is the y-intercept.)
Parallel vs. Perpendicular
Parallel lines have identical slopes. In other words, for two parallel lines, m1 = m2
Perpendicular lines have slopes that are negative reciprocals of one another. In other words, for two perpendicular lines m2 = -1/m1
Perpendicular lines have slopes that are negative reciprocals of one another. In other words, for two perpendicular lines m2 = -1/m1
Functions: Domain, Range, and Zero
Functions describe a dependent relationship between quantities.
One form of a function is f(x) = mx + k. You can state this as "the quantity f is dependent on the value of x."
The Domain of a function is the set of values of x for which the function is defined.
The Range of a function is the set of possible output values.
The zero of a function is the value of x for which the output of the function = 0.
One form of a function is f(x) = mx + k. You can state this as "the quantity f is dependent on the value of x."
The Domain of a function is the set of values of x for which the function is defined.
The Range of a function is the set of possible output values.
The zero of a function is the value of x for which the output of the function = 0.
Real Numbers
Imaginary Numbers
Complex Numbers
Complex numbers have the form a + bi.
- When b = 0, the complex number is Real.
- When b ≠ 0, the complex number is Imaginary.
- When a = 0, the complex number is a Pure Imaginary number.
Complex Conjugates
a - bi is the Complex Conjugate of a + bi (and a + bi is the Complex Conjugate of a - bi.)
- The sum of Complex Conjugates is a real rumber.
- The product of Complex Conjugates is a nonnegative real number.
- For every complex number
the corresponding Complex Conjugate is
Quadratic Equations
Methods for finding the solution/root:
- Factoring
- Completing the Square
- Quadratic Formula
Graphing Quadratic Equations
Step 1: Find y-intercept. Let x = 0, and solve for y.
Step 2: Find x-intercepts. Let y = 0 and solve for x. (Check for extraneous roots!!)
Step 3: Find axis of symmetry, which is equal to -b/2a
Step 4: Find vertex. Let x = -b/2a and solve for y.
Step 1: Find y-intercept. Let x = 0, and solve for y.
Step 2: Find x-intercepts. Let y = 0 and solve for x. (Check for extraneous roots!!)
Step 3: Find axis of symmetry, which is equal to -b/2a
Step 4: Find vertex. Let x = -b/2a and solve for y.
Polynomial Functions
This is an example of a "quartic" function, which has 5 terms:
- ax^4 is the "lead term" and determines the degree of the polynomial
- a, b, c, d are coefficients
- e is the constant term.