Quadratic
Relation, Expression and Equations
About Quadratics
So get ready with your pens and notebooks :D
Table of Context
- Introducing the Parabola.
- Analyzing quadratics
- Graphing from vertex form
- Transformation.
- Finding the equation given vertex
- Graphing from factored forming
- Expanding
- Common factoring
- Factoring by grouping
- Factoring simple trinomials
- Factoring complex trinomials
- Factoring special trinomials
- Video on Factoring
- Completing the Square
- Solving by isolating x
- Discriminant
- Quadratic Formula
- Shape problems
- Revenue problems
- Motion problems
- Consecutive intergers
- Reflection
- Links between topics
Introducing....The Parabola
Main ideas:
- Parabolas can open up or down
- The zero of a parabola is where he graph crosses the x-axis
- Zeroes can also be called x intercepts or roots
- The axis of symmetry divides the parabola into equal halves
- The vertex of a parabola is the point where the axis of symmetry and the parabola meet.
- The optimal value is the value of the y co- ordinate of the vertex
- The y intercept of a parabola is where the graph crosses the y axis.
Analyzing Quadratics.
For example:
Graphing from vertex form
y = a(x - h )2+ k
In order to graph from vertex form, you need to know what each variable represent:
For example:
For example:
- The original step pattern of a parabola is over one up one, over two up four.
- The vertex decides where we start the step pattern.
- The 'a' value decides if the step pattern changes (it multiples the vertex part of the step pattern
Graphing transformations of Parabolas
Finding the equation given vertex
Graphing from factored form
y=a(x-r)(x-s)
You need to know how each variable affects the graph before we start learning how to graph
1. a - causes the vertical stretch or compression. If a>1 it is a vertical stretch. If -1<a>1 its is a vertical compression.
- if the parabola face up or down. If a is negative, the parabola face down. If a is positive, the parabola faces up.
2. r - first x-intercept
3. s - second x-intercept
Below is a YouTube video to help you understand how to graph a parabola from factored form:
Expanding
The example below explains how expand an equation
Common Factoring
When you find the factors of two or more numbers, and then find some factors are the same ("common"), then they are the "common factors".'
Factoring by grouping
Factoring Simple Trinomials
Equation- x2 + bx + c
Step 1: Find multiples of x2 and c
Step 2: Form two brackets
Step 3: Insert the multiples of x2 and c in both of the brackets
Step 4: Expand the brackets to check if you got the right answer
Below is an example of simple trinomials question.
Factoring Complex Trinomials
Equation - ax2 + bx + c
Step 1: Find multiples of ax2 and c
Step 2: Form two brackets
Step 3: Insert the multiples of ax2 and c in both of the brackets
Step 4: Expand the brackets to check if you got the right answer
Below is an example of complex trinomials
Factoring Special Trinomials
Step 1: Find the multiples of a and c
Step 2: Multiply the numbers by 2 to get the middle number
Step 3: Insert the multiples of the squares in the equation above
Different of squares - ( x + z ) ( x - z )
Step 1: Find the multiples of x and c
Step 2: Insert both the numbers you got in the brackets one + and the other -
It is very important to know the difference between them. Below is an example of this equations
Video on Factoring
Completing the square
Example:
- Remove the common factor form x2 and x term -coefficient
- Find the constant that must be added and subtracted to create a perfect square
- Group the 3 terms that form the perfect square. (move the subtracted value outside the bracket b multiplying it by the common factor first)
- Factor the perfect square and collect like terms
Solving by isolating for x
Below is a step by step example that shows you how to answer questions like this.
Discriminant
Discriminant - b2 - 4ac
It quickly tells you the number of real roots, or in other words, the number of x-intercepts, associated with a quadratic equation.
D>0 - 2
D<0- 0
D=0- 1
Below is an example of how to find a discriminant
Quadratic formula
A new formula is introduced which is:
Below is an example of a quadratic formula question
Shape problems
The questions mostly asked is to find the area, perimeter and missing sides.
Below is an example about a right triangle
Revenue Problems
Below is an example of a revenue problem
Motion Problems
Example:
Consecutive Integers
Below is an example of a consecutive integers
REFLECTION
Links between topics in Quadratics
Factoring connects to Graphing
- Factoring helps you understand equations such as complex trinomials and helps you know what each variables does while graphing
- It also helps you know what the x-intercepts are and where the vertex is by calculating
Standard form connects to Vertex form
- Both topics are linked back together through the topic of Completing the square
- You convert standard form to vertex form in order to get the vetex
Discriminant connects to Quadratic Formula
- The discriminant equation goes back to half of the quadratic equation
Simple trinomials connects to Complex trinomials
- Simple and Complex have the same equation except for how the a-value is equal to 1 in simple but in complex, the a-value equals to more than 1