Quadratic Relations
By: Bilal Baig
Table of contents
Types of Forms
Vertex Form
Factored Form
Standard Form
Axis of symmetry
Optimal Value
Transformations
X-intercepts or zeroes
Step pattern
Graphing parabolas
Factored Form y= a (x-r)(x-s)
Zeroes or x intercepts
Axis of symmetry
Optimal value
Standard form y= ax²+ bx + c
Zeroes
Axis of symmetry
Optimal Value
Completing the square to turn to vertex form
Common factoring
Factoring to turn to factored form
Factoring to complex trinomial
Perfect squares
Difference of Squares
Word Problems
Reflection
Types of forms
- Vertex form - y = a(x-h)²+k
- Factored form - a(x-r)(x-s)
- Standard form - y = ax² + bx + c
Vertex form
Axis of symmetry
The axis of symmetry is the center point of the parabola on the x axis. In vertex form, the axis of symmetry is the opposite symbol of the h value. For example in the equation y= (x+5)² - 3 the axis of symmetry would be -5 because h is a negative so when you put two negatives together they become a positive.
Optimal Value
The optimal value is the maximum or minimum point of the parabola on the y axis. In vertex form the optimal value is K value. In the equation y= (x-5)² - 3 the optimal value would be -3 and minimum.
Transformations
The (-h) moves the parabola left or right. The parabola moves left if (-h) is a positive like (+5) and it moves right if it is a negative like (-5).
The (K) moves the parabola up or down. The parabola moves up if K is a positive and it moves down if K is a negative.
The (A) stretches the parabola. If the A is negative the parabola opens down, if A is a positive it opens up
Zeroes
For example:
Step Pattern
Graphing parabolas
Factored Form
Zeroes (X-intercepts)
Axis of Symmetry
To find the Axis of Symmetry you must solve for x by setting y=0 and solve each bracket separately then put the x intercepts together and divide by 2.
Optimal Value
Standard Form
Zeroes
Axis of Symmetry
Completing the square
- First you must put ax + bx in brackets, then common factor the numbers and place the factor in front of the brackets.
- After that you do (b/2)² and add it to bx, but since you can't just add to an equation you also have to subtract it too.
- Next you'll notice in the bracket you have a perfect square (x² + 4x + 4) so now you square root the x² and 4 leaving you with x and 2. Then you put the x and 2 together in brackets and square it.
- After that you multiply the - 5 with the - 4.
- Then add the + 20 to +15 and you have your K value which is also the optimal value.
Optimal Value
Common factoring
Factoring to Factored Form
Factoring Simple Trinomials
Factoring complex Trinomials
Perfect Squares
Difference of Squares
Word Problem
Motion problem
1.a) What is the max height? When does it happen?
1.b) When will the ball hit the ground?
1.c) Sketch a graph.