Quadratic Relations Website
Everything you need to know about Grade 10 Quadratics!
Made by Kamran Samra. Grade 10. Novemeber 2014
What is Quadratics anyway? Give an example
For example, Quadratics can be used to graph the flight path of an object, for example, in football when you throw the football it makes a U shape and we can use quadratics to ensure that the ball will go in the direction and location you want it to go in. You can also use quadratics when it comes to business, you can use them in business meetings with charts when you want to make the highest amount of money possible and the highest point (the vertex) will be that mark. Quadratics are everywhere!
Table of Contents
Introduction to Quadratics
- What is a parabola? (Basic Terminology)
- Second Differences
Vertex Form y=a(x-h)²+k
- Effects of Vertex form
- Axis of symmetry
- Optimal Value (y=k)
- Transformations
- Step Pattern
- X-intercepts or zeros (Subbing in y=0 and solving)
- Graphing Vertex Form
Factored Form y=a(x-r)(x-s)
- Zeroes or x-intercepts (r and s)
- Axis of symmetry x=(r+s)/2
- Optimal Value (Subbing in)
Standard Form y=ax²+bx+c
- Zeroes (Quadratic Formula)
- Axis of Symmetry (-b/2a)
- Optimal Value (Subbing in)
- Completing the square to turn to vertex form
- Factoring to turn to factored form
Factoring
- Common Factoring
- Simple Trinomial
- Complex Trinomial
- Perfect Squares
- Difference of Squares
- Factoring by Grouping
Welcome to the world of Quadratics
What is a Parabola?
Vertex- where the optimal value and the axis of symmetry meet. (The highest and lowest part of the line)
Axis of symmetry- the half line of the curve (vertical line that splits the parabola in half)
Optimal Value- the highest or lowest point of the parabola (determined by the vertex)
Y-intercept- the point when the curve crosses the y-axis ( all numbers above optimal value)
X-intercepts/Roots/Zeros- the points where the curves cross the x-axis (all real numbers)
Second Differences
Now that we have a basic overview of Quadratics, Lets Begin with Vertex Form
Effects of Vertex Form. What is it?
Axis of Symmetry
Optimal Value
Transformations. Each variable represents a transformation
The (-h) value moves the vertex of the parabola left or right
When (-h) is negative move right and when its positive move left.
The (k) value moves the vertex of the parabola up or down
When (k) is negative move downwards and when its positive move upwards.
The (a) stretches the parabola and if the
If (a) is positive the parabola opens upwards, if negative it opens downwards.
Vertex= (h,k)
Step Pattern
x-Intercepts (y=0)
To find the x-intercepts in Vertex Form, we must sub y=0.
Example: y=4(x-3)²-1
y=0.
0 = 4(x² - 6x + 9)-1
0 = 4x² - 24x + 35
0 = (2x-5)(2x-7)
x = 5/2 and x = 7/2
x-intercepts (2.5,0) (3.5,0)
Graphing Vertex Form y=a(x-h)²+k
Step 2: Find the y-intercept (Let x=0 and solve for y)
Step 3: Find the x-intercepts (Let y=0 and solve for x)
Step 4: Graph the parabola using all 3 points (vertex, x-int, y-int)
Please see the video attached for clarification.
Factored Form y=a(x-r)(x-s)
Zeroes or x-intercepts (r and s)
A zero of a parabola is another name for the x- intercepts.
in order to find the x- intercepts you must set y=0
Example: y=-(x-2)(x+4) Each bracket represents a x-intercept.
0=(x-2) Move the -2 to the opposite side of the equal sign to isolate x and change it to +2
x=+2 (2,0)
0=(x+4) Move the +4 to the opposite side of the equal sign to isolate x and change it to -4
x=-4 (-4,0)
The zeroes (x-intercepts) are (2,0) (r) and (-4,0) (s) in the equation y=-(x-2)(x+4)
Axis of Symmetry x=(r+s)/2
To find the axis of symmetry, we must get our two zeros from before (2 and -4), add them and then divide them by 2. Using the following formula x=(r+s)/2
-4+2=-2 Add both the zeroes together.
-2/2=1 Divide the number by two.
Therefore the axis of symmetry is -1.
The image to the right shows what an Axis of Symmetry is (blue line)
Finding the Optimal Value (Vertex) using Substitution
To find the Optimal Value you have to sub in the axis of symmetry into the equation we had from the beginning. y=-(x-2)(x+4)
y=-(x-2)(x+4)
x=-1
y=-(-1-2)(-1+4)
y=-(-3)(3) Expand the brackets with the a value (-1)
y=9
Vertex= (-1,9)
You can now get your Optimal Value and x-intercepts and plot them on the graph to create your parabola
Please refer to the attached video for clarification on how to graph using Factored Form
Standard Form y=ax²+bx+c
Axis of Symmetry (-b/2a)
Example: 5x²+6x+1. b=6 a=5
-(6)/2(5)
-6/10
=-0.6
Therefore the axis of symmetry (-0.6,0)
Zeroes (Quadratic Formula) x = -b ± √b² - 4ac / 2a
Identify the coefficients (a=5 b=6 c=1)
Now sub the values into the formula
x=-(6)± √(6)²-4(5)(1) / 2(5)
x=-6± √(36)-(20) /10
-6± √(16) /10
(-6±4) /10
-6+4/10=-0.2 (x-intercept (-0.2,0)
Or
-6-4/10=-1 x-intercept (-1,0)
Two x-intercepts are are (-0.2,0) and (-1,0)
NOTE: You cannot take the square root of a negative number. (No x-intercepts)
Discriminates (D)
It shows how many solutions (x-intercepts) the equation has
Note:
If the Discriminate is negative, there are no solutions to the equations
If the Discriminate is positive, there are 2 solutions to the equation
If the Discriminate is zero, there is 1 solution to the equation.
Completing the Square to Turn into Vertex Form
Example: y=2x²+8x+5
First, Group like terms like (x² and x terms) together by putting brackets around them.
y=(2x²+8x)+5
Second, Common factor (ONLY THE CONSTANT TERMS)
y=2(x²=4x)+5
Third, Complete the square inside the bracket
y=2((x²+4x+4) -4) +5
(4/2)² ^ ^ Note: You need to subtract 4 so you don't make any changes to the equation!
Fourth, Write the trinomials a binomial squared . We must also expand into the brackets.
y=2 (x+2)² -8+5
Vertex Form: y=2(x+2)²-3 The Vertex (h,k) is -(2,-3)
Please use the video below for reference
Factoring
Common Factoring
Example: 8x+6
First, we have to find the GCF (Greatest Common Factor) which is 2
Divide 8 and 6 by 2.
Your new answer should be
=2(4x+3)
Please refer to the example and video posted below
Simple Trinomials x²+bx+c
2 numbers that add up to the b value
2 numbers that multiply together to give the c value
Example: x²-7x-18
=(x-_)(x+_)
So what 2 numbers that are added together equal -18 and added together equal -7
-9x2=-18
-9+2=-7
Therefore the answer is (x-9)(x+2)
For any further questions please refer to the video below
Complex Trinomial x²+bx+c
There are two ways to factor Complex Trinomials, the first way is:
Decomposition
Example: 5x²+12x+4
The first and last value multiply together (5x4=20) We have to find what two numbers multiply together that add to 20 and add together to get 12 (10 and 2)
Now we replace 10x and 2x into 12x value and our trinomial now looks like
(5x²+10x)+(2x+4)
Now we have to find the GCF for the first and second brackets and divide them. First Pair=5x Second Pair=2
5x(x+2)+2(x+2)
Therefore the answer is (5x+2) (x+2)
Trial and Error
Here, we have to guess and check and keep substituting numbers until we get the correct answer.
Example: 4x²+12x+9
First we start off with two brackets, we know that we can use (2x+__)(2x+__) that will get us 4x². Now we have to substitute numbers in that will get us 4x²+12x+9.
(2x+3)(2x+3) Expanded: 4x²+6x+6x+9 (Collect like terms and we get 4x²+12x+9)
For any further questions please refer to the video I prepared.
Perfect Squares and Difference of Squares
Perfect of Squares include 1,4,9,16,25,36,49,64,81,100 etc.
First, the first number and third number can be square rooted
Example: 4x²+12x+9
√4x²=2x
√9 =3
(2x+3)²
Check your answer
(2x)(3)(2)=12x
Difference of squares
Two numbers in the expression where both numbers can be square rooted and the second number has to be negative.
Sample Assesments
Reflection
Word Problems
Motion Problems
Area Problems
Calculators are sold to students for 20 dollars each. 300 students are willing to buy them at that price. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. What selling price will produce the maximum revenue and what will the maximum revenue be?