By: Simran Heer
Table of Contents
- Second Differences
- Properties of Parabolas
- Transformations In Vertex Form
Types of Equations:
- Factored Form: a(x-r)(x-s)
- Standard Form: ax² + bx + c
- Vertex Form: a(x-h)² + k
- 2 examples
Properties of Parabolas
- Vertex: where the graph changes direction (x, y)
- Axis of Symmetry: the line which passes when x=0, and divides the parabola into two equal parts
- Optimal Value: the y coordinate, either the highest (max) or lowest (min) point
- X-intercepts: the points which cross the x axis. There could be 2 solutions, 1 solution, or no solution.
- Y-intercepts: when the graph crosses the y axis
Types of Equations
Factored Form: a(x-r)(x-s)
To graph a parabola in factored form, you need to:
Find the Axis Of Symmetry:
- Find the zeros
- Solve by adding both zeros, then dividing by 2. This will give you the AOS
How to find the Vertex:
- Substitute the value of the axis of symmetry in the factored equation. Solve to find y.
Expanding Factored to Standard
You use the method of expanding when you want to go from factored to standard form. You multiply each term in the brackets, and then collect the like terms.
Simple Factoring: x² + bx + c = (x + r)(x + s)
(a - b)(a + b), and a polynomial of the form a² x² +- 2 a b x + b is a perfect square and can be factored as (a x +- b)².
Factor by Grouping
Vertex Form: a(x - h) + k
- The (-h) moves the vertex of the parabola left or right , when the (h) is negative it moves to the right and when its positive it moves to the left.
- The (k) moves the vertex of the parabola up or down, when the (k) is negative it moves down and moves up when its positive.
- The (a) vertical stretch or compression. The parabola is reflected upon the axis if a the (a) is negative.
When solving for the x-intercepts set y=0 and solve the equation. There could be 2 solutions, 1 solution, and no solution.
The vertex is given by the equation, where (h, k) is the vertex.
Standard Form: ax² + bx + c
- (a) gives the shape and direction of opening of the parabola
- (c) is the y-intercept
You can solve for the x-intercepts using factoring, completing the square, or the quadratic formula.
The Quadratic Formula solves for the x-intercepts
Find the X-Intercepts:
- Use the quadratic formula to get the zeros.
- Add the two solutions together and then divide by 2 to get the axis of symmetry.
- Substitute the (x) value you solved for above, and solve for y.
- The axis symmetry will be the (x) value in the vertex
- Vertex: (h,k) h=x value, k=y value
Link Between Equations and Graphs
Discriminant and X-intercepts: Discriminant's tells you how many solutions you have to an equation or parabola. This helps us find the exact number of x-intercepts.