Sequences

Things to Know - Arithmetic Sequence (t_n = d (n - 1) + t₁), Quadratic Sequence, and Power Sequence (p.11 #30). Common Difference, infinite Sequence, and finite sequence.

Type of Sequence	When Common	Degree	Equation
Linear	D₁ is constant	1	t_n= an + b or t_n= t₁+ d(n-1)
Quadratic	D₂ is constant	2	t_n= an² + bn + c
Cubic	D₃ is constant	3	t_n= an³ + bn² + cn + d
Quartic	D₄ is constant	4	t_n= an⁴ + bn³ + cn² + dn +e
Quintic	D₅ is constant	5	t_n= an⁵ + bn⁴ + cn³ + dn² +en + f
Geometric	Common Ratio	n	t_n = t₁(r)^(n-1)

Completing the Square and Transformational Form

A procedure called “completing the square” is used to change from general to transformational form of an equation.

Step 1: Move the constant at the end to the other side of the equation.	Example:
Step 2: Divide everything by the coefficient of the x²-term.
Step 3: Take half the coefficient of the x-term and square it. Add this square to both sides of the equation
Step 4: Factor the right side of the equation into a perfect square
Step 5: Factor out the coefficient of the y-term to get the stretch and horizontal translation.

The Quadratic Formula and Decomposition

-The quadratic formula can be used to find the “roots” or x-intercepts of a parabola. This is where the curve crosses the x-axis and y is equal to zero. There are three ways to find the roots of an equation. The simplest way is to factor the equation, set it equal to zero and solve for x. If the function does not factor easily, you may try factoring using decomposition. If it really looks nasty, dive right into the quadratic formula.

Finding Roots using Decomposition Example

Step 1: Multiply the coefficients of the first and last term together. Find factors of this product that add to the coefficient of the middle term. Product of first and last term: (2)(-6) = -12 Factors: (-1, 12), (1, -12), (-2, 6), (2, -6) (-3, 4) and (3, -4)
Step 2: Split the middle term into the two factors you found in the last step
Step 3: Group the first and last two terms together
Step 4: Factor out of the groups
Step 5: Factor the common term out of each group
Step 6: Set each factor equal to zero to solve for x

The Quadratic Formula: when

Finding Roots with the Quadratic Formula Example

Step 1: let y = 0
Step 2: Enter a, b and c coefficients into the Quadratic Formula
Step 3: Simplify
Step 4: Solve for x	and

What a Function’s Roots Tell you about Its Graph - A quadratic function has two roots and you can find out what type of roots they are by using the discriminant. The discriminant is the part of the quadratic formula.

A Quadratic function with...	two real roots	a double root (two roots of the same value)	two non-real roots
has a discriminant...	positive discriminant	zero discriminant	negative discriminant
looks like this ...
and has this many x-intercepts	two x-intercepts	one x-intercept. I call this one a “skimmer” because it just touches the x-axis and then flies away	zero x-intercepts

Finding the Formula of a Parabola

If you are given the vertex of a parabola and any other point on the curve, you can determine the formula for the parabola using the transformation for of the quadratic. The vertex is (h, k) and the other point is (x, y). When you plug the numbers into the equation, you can solve for a (the stretch).