Quadratics
By Ramanveer Brar
Table of Contents
Linear or Quadratic
Parabolas
Graphing Parabolas Video
Factoring
Types of Forms
All About Vertex Form
The Axis of Symmetry and Optimal Value
Transformations
Zeros & X-intercepts
Step pattern
All About Factored Form
Zeros & X-intercepts
Axis of Symmetry
Optimal Value
All About Standard Form
Zeros
Axis of Symmetry
Optimal Value
Completing the Square to Turn to Vertex Form
Factoring to Turn to Factored Form
- Common
- Simple Trinomial
- Complex Trinomial
- Perfect Squares
- Difference of Squares
Word Problems
Making Connections
Reflection
Vocabulary
Curve of best fit: a smooth curve drawn to approximate the general path or trend in a scatter plot.
Parabola: the graph of a quadratic relation, which is U-shaped and symmetrical
Vertex: The point on a parabola where the curve changes direction. The maximum point if the parabola opens down. The minimum point if the parabola opens up.
Axis of symmetry: the line that divides a figure into two congruent parts.
Zero: a value of x for which a relation has a value of 0. Corresponds to an x-intercept of the graph of the relation.
x-intercept: the x-coordinate of the point where a line or curve crosses the x-axis. At this point y=0
Perfect square trinomial: a trinomial that is the result of squaring a binomial.
Difference of squares: an expression that involves the subtraction of two squares.
Linear or Quadratic
Parabolas
- Parabolas can open up or down
- The zero of a parabola is where the graph crosses the x-axis
- "Zeros" can also be called "x-intercepts" or "roots"
- The axis of symmetry divides the parabola into two halves
- The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value.
- The optimal value is the y co-ordinate of the vertex
- The y-intercept of a parabola is where the graph crosses the y-axis
To graph parabolas you can use Desmos. It is a quick substitute to drawing a parabola onto graph paper.
Factoring
Types of Forms
Vertex form - a(x-h)²+k
Standard form - ax²+bx+c
Vertex Form
Axis of Symmetry (x=h)
Optimal Value (y=k)
Transformations
Orientation
If the a value is greater than zero the parabola opens up.
If the a value is less than zero the parabola opens down.
Shape
If the a value is zero the parabola is vertically compressed.
If the a value is greater or less than one the parabola is vertically stretched.
Zeros & x-intercepts
Step Pattern
The step pattern is the rate at which the parabola goes up/down.
Ex. up 1, up 3, up 5, up 7, each time it goes up more but to the side by 1. This is just one example.
Factored Form
y = ( x - 3 ) ( x - 4 )
x - 3 = 0 x - 4 = 0
x = 3 x = 4
Axis of Symmetry
x = 3 + 4 / 2
x = 3.5
Optimal Value
y = ( x - 3 ) ( x - 4 )
y = ( 3.5 - 3 ) ( 3.5 - 4 )
y = 12.25 - 14 - 10.5 -12
y = -24.25