Quadratics
By: Sonali Thakkar
Table of Contents
- Features
- Parabolas
- Ways to represent Parabola's
- Vertex Form
- Factored Form
- Standard Form
Vertex form
- Investigating
- Graphing
- Transformations
- How to find a equation using a graph
Mini Test #1
Expanding and Factoring:
- Multiplying Binomials
- Special Products
- Common Factoring
- Factoring simple Trinomials
- Factoring Complex Trinomials
- Differences of square Trinomials
- How to find a equation using a graph
Mini Test #2
- Finding the Maxima and Minima
- Graphing using x-Intercept
- The Quadratic Formula
Test #3
Word Problems
Introduction to Quadratics
Key Features of a Quadratic Realtion
Description:
- the opening of the parabola is up
- the axis of symmetry is the x value of the vertex
- the optimal value is the y value of the vertex
- the zeros are when the parabola touches or goes through the x axis
- y- intercept is when the the parabola goes through the y axis
- the vertex is the minimum or maximum point of the parabola
Ways To Represent Parabolas
Table of Values
Graph
Equation
3 Types of Equations
- Standard Form
- Vertex Form
- Factored Form
Standard Form
Vertex Form
where h and k represent the vertex. h represents the x value of the vertex. If h is positive in the equation, it is negative on the graph. If h is negative in the equation then, it is positive on the graph.
k represents the y value of the vertex. The value of k does not have any special rules, as you see it on the equation is where you place it on the graph.
Factored Form
Vertex Form
Investigation
How to Graph with Vertex Form
Basic Transformations
The four transformation are:
- if your parabola has an opening up or down
- if your parabola is vertically stretches or compressed
- how many units is it transforming left or right
- how many units it is transforming up or down
For example:
y= -2(x-4)-5
The transformations for that equation would be:
- it has opening down
- the parabola has a vertical stretch of 2
- the parabola if transformed 4 units to the right
- the parabola is transformed 5 units down
How to Find a Equation Using a Graph
Unit Test 1
Expanding and Factoring
Expanding Binomials
For example, (3x+2)(3x-3)
To expand this binomial, you have to:
- First, you multiply the outer term with inner term. so you would multiply (3x)(3x)= 9x^2
- Second, you multiply the outer term with the inner term. (3x)(-3)= -9x
- Third, you multiply the two inner terms. (2)(3x)= 6x
- Fourth, you multiply the inner term with the outer term. (2)(-3)= -6
- Fifth, you group the like terms.
9x^2-3x-6
- Finally, you have you standard form equation 9x^2-3x-6
Special Products
(x+4)^2
x is the value a and 4 is the value of b, so now you plug that into the formula:
x^2+ 2(x)(4)+ 4^2
x^2+8x+16
Common Factoring
1) Monomial Common Factor
2) Binomial Common Factor
3) Factor by Grouping
Monomial Common Factor:
i) Find the GCF of coefficients and variables
ii) Divide each term by GCF
Examples:
5c+10d
- the GCF is 5 therefore you divide both terms by 5.
- you would rewrite that as
5(c+2d)
8x^2-7x
- the GCF is x therefore you divide both terms by x.
- you would rewrite this as
Binomial Common Factor:
i) if there are two binomials that are exactly same, consider that as a binomial common factor.
Examples:
- x(x-2)+2(x-2)
- x-2 are the same therefore they are the common factor and you would rewrite it as
(x-2)(x+2)
- the common binomial goes first and what is left comes after.
Factor by Grouping:
i) factor groups of two terms with a common factor to produce a binomial common factor.
Example:
- ax+ay+2x+2y -ax and ay have the common factor a and 2x and 2y have the common factor 2.
- (ax+ay)+(2x+2y) - then you add brackets around common terms.
- a(x+y)+2(x+y) - then you take the common factor outside the brackets. At this you know you are correct if both of the terms inside the brackets are the same.
- then you set them into a binomial common factor
Simple Trinomial Factoring
x^2+bx+c = (x+r)(x+s)
There are 3 steps to factor a simple trinomial.
Step 1: Find the product and sum
- Find two numbers whose product is c
- Find two numbers whose sum is b
For example,
x^2+12x+27
- two numbers whose product is 27 and also have the sum 12 are 3 and 9.
Step 2: Look at the signs of b and c in the given expression
- If b and c are positive, both r and s are positive.
Both b and c are positive therefore, r and s are positive and the answer is
(x+4)(x+3)
- If b is negative and c is positive, both r and s are negative.
b is negative and c is positive therefore, both r and s are negative.
(x-28)(x-1)
Step 3:
- If c is negative, one of r or s is negative.
For example:
x^2+3X-18
c is negative therefore either one of r or s is negative.
(x-3)(x+6)
Factoring Complex Trinomials
Step 2: To factor ax^2+bx+c, find two integers whose product is ac and whose sum is b.
Step 3: The check up the middle term and factor by grouping
Example: 3x^2 +8x+4
- Two integers whose product is (3)(4) and sum is 8
- Break up the middle term
- Factor by grouping
(3x+2)(x+2)
Differences of Squares
Example: x^2-16
x^2-4^2
(x+4)(x-4)
Unit Test 2
Finding the Maxima and Minima
Completing the squares turns a standard form equation (x^2+bx+c) into a vertex form equation (y=a(x+k)^2-h).
Step one: you add brackets around your a and b
Step 2: divide your b value by 2 and square your answer
Step 3: put a positive square in the bracket and a negative square outside the bracket
Step 4: you now have a perfect square trinomial in the brackets, so we turn that into a binomial.
Step 5: solve the values outside the bracket
For example:
y=x^2+8x+3
y=(x^2+8x)+3
y=(x^2+8x+16)-16+3
y=(x+4)^2-16+3
y=(x+4)^2-13
Therefore, the vertex of this equation is (4,-13).
Graphing Using the x-intercepts
y=x^2-6x+8
-4,-2
y=x^2-4x-2x+8
y=(x^2-4x)(-2x+8)
y=x(x-4)-2(x-4)
y= (x-4)(x-2)
x-4=0 x-2=0
x=4 x=2
(4,0) (2,0)
Therefore, the x-intercepts are (4,0) and (2,0)
4+2/2=6/2=3
the AOS is 3. Now you plug in your x value.
y= (x-4)(x-2)
y= (3-4)(3-2)
y=(-1)(1)
y=-1
Therefore the vertex is (3,-1)
Now we can graph this because we know our x-intercepts, our vertex, and the y-intercept. The y-intercept is the value of c in the original equation.
The Quadratic Formula
- When given your standard form equation you determine you a value, your b value and c value so there is no confusion.
- You then plug in all of you variables.
- First, you square your b and multiply your -4ac
- Second, multiply your 2a
- Third, solve inside the square root
- Fourth, square root that answer
- Now the equation goes two ways; one you add your -b with your square root divided by your 2a; two you subtract your -b with your square root and divide it with your 2a
- The two answers you get are your x-intercepts.
Revenue Word Problems
a) Create an algebraic equation to model Bryce's total sales revenue
Let (18.50+0.50x) represent the cost
Let (216-8x) represent the total number of people
R= (18.50+0.50x)(216-8x)
b) Determine the maximum revenue and the price at which this maximum revenue will occur
R= (18.50+0.50x)(216-8x)
18.50+0.50x=0 216-8x=0
0.50x= -18.50/0.50 -8x= -216/-8
x= -37 x= 27
-37+27/2= -10/2 = -5
R= (18.50+0.50x)(216-8x)
R= (18.50+0.50(-5))(216-8(-5))
R=(18.50-2.5)(216+40)
R=(16)(256)
R= 4096
Therefore the maximum Revenue $4096 will occur at the maximum price of $16