Quadratics
By: Jaspreet Chahal
Table of Contents
Introduction
Analyzing Quadratics
Quadratic Terms
Step Pattern
Parabola Transformations
Types of Equations
Quadratic Relations
Factored Form
Vertex Form
Quadratic Expressions
Expanding
Common Factors
Factoring Simple Trinomials
Factoring Complex Trinomials
Perfect Squares
Difference of Squares
Quadratic Equations
Quadratic Equations
Quadratic Formula
Quadratic Equation Problems
Geometry
Revenue
Reflection
Introduction
Analyzing Quadratics (Using Table of Values)
Quadratics. Before we move on, we need to know how to spot a quadratic relation. You will be able to spot a quadratic relation if the second differences are constant and the first differences are not.
Quadratic Terms
Here are some terms and a diagram that will help you understand parabolas more:
Vertex: The maximum or minimum point on the graph. It is the point where the graph changes direction.
Optimal Value (Maximum or Minimum Value): The highest/lowest point on the parabola.
Axis of Symmetry: A line which that goes down the middle of the parabola.
Y-intercept: The point where your parabola touches the y-axis.
X-intercepts: The point where your parabola touches the x-axis.
Zeroes/Roots: The point where the parabola will cross the x-axis.
Step Pattern
For example, if your equation was y=x², then your base step pattern would be to start from the vertex and move over 1 and up 1, then over 2 and up 4, and so on. If your A was a number other than 1, then it would affect the vertical stretch of the parabola. The step pattern would change. Now you would move over 1, then multiply move up 1 (the vertical stretch), with the number that is your A, and then move that many numbers up to get your second point.
Parabola Transformations
a = stretches the parabola. It multiplies the vertical stretch. (When a is negative, the parabola opens down, when a is positive, the parabola opens up)
h = moves the vertex of the parabola left or right. (When h is negative, the vertex moves right, when h is positive, the vertex moves left)
k = moves the vertex of the parabola up or down. (When k is negative, the vertex moves up, when k is positive, the vertex moves down)
Types of Equations
1. Standard Form y=ax²+bx+c
2. Factored Form y=a(x-r)(x-s)
3. Vertex Form y=a(x+h²)+k
Quadratic Relations
Factored Form
Equation: y=a(x-r)(x-s)
In the example I will teach you how to make a parabola using the factored form equation.
Before that, here are the terms that are going to be used:
Zeroes: found by setting each factor equal to zero
Axis of Symmetry: midpoint of zeroes
Optimal Value: found by subbing axis of symmetry value into the equation
Vertex Form
Equation: y=a(x+h²)+k
Finding the Equation Given the Vertex
With the vertex) and the other point you were given, you must sub that information into the equation. With this equation you must now find the value of a. When you find the value of a, you will be able to write the complete equation.
Quadratic Expressions
Expanding
Expanding is very simple. In the next 2 examples I will show you how to expand with algebra tiles, and without algebra tiles.
Common Factoring
Factoring Simple Trinomials
Factoring: ax²+bx+c
b,c= different numbers
a=a variable, in this case, 1
In the next video I will show you how to factor simple trinomials.
Factoring Complex Trinomials
Factoring: ax²+bx+c
b,c= different numbers
a= a variable, in this case, any number, negative or positive
a multiplied by c should equal the product, and a+c should equal b.
In the example I will show you how to factor complex trinomials.
Perfect Squares
Here are the perfect square equations:
(a²+b²) = (a+b) (a+b)
(a+b)² = a²+2ab+b²
Here is an example of solving a perfect square.
Difference of Squares
Here are the difference of squares equation:
(a²-b²) = (a+b) (a-b)
(a-b)² = a²-2ab+b²
Here is an example of solving a difference of squares.
Quadratic Equations
Quadratic Equations
Quadratic Formula
Quadratic Equation Problems
Word Problem - Geometry
Word Problem - Revenue
Reflection
Overall, the quadratics unit was very long and a bit challenging for me. I did not do so well on mini test 1, I some what improved on mini test 2, and I hope to do well on mini test 3.
We do not realize that we see parabolas everywhere, but now that I have learned about it, every time I see a parabola I ask myself, did they use quadratics to build that?!
And it's obvious that they did, and it's amazing how much work they had to put in to build it.
Throughout this unit I have learned many different ways on how to make a parabola and I hope to use these skills next year.