Quadratics
By : Azan B.
Table of contents
Introduction
What is it
The Algebra Of Quadratics
Quadratic Function
Analyzing Quadratics (Second Differences)
The Parabola
Transformations
Types of Equations
Factored Form a(x-r)(x-s)
Vertex Form a(x-h)²+k
Standard Form ax²+bx+c
Vertex form
Finding an equation when only given the vertex
Graphing vertex form
Factored Form
Factors and zeroes
multiplying out brackets and quadratic expressions
Common factoring
Factoring simple trinomials
Factoring complex trinomials
Perfect square
Factoring by grouping
Graphing factored form
Standard form
Completing the square
Quadratic formula
Graphing from standard form
Word Problems
Revenue
Numeric
Reflection
INTRODUCTION
What is it?
The algebra of quadratics
Quadratic Function
Analyzing quadratics (Second Differences)
Parabola
vertex- is the highest or lowest point of the parabola
Line of symmetry- The very middle of the parabola through the x axes
x-intercept - Is where the parabola crosses the x axes
y-intercept - is where the parabola crosses the y axes
Graphical Transformation Of Functions
Scaling The Quadratic
Types of equations
Vertex form: a(x-h)²+k
Factored form : a(x-r)(x-s)
Standard form: ax²+bx+c
Factored form
You have to multiply everything within the brackets by each other before multiplying it by "a".
(x)(x) (x)(s) (r)(x) (r)(s)
After multiplying these you then simplify and then multiply it all by the a value
This is called expanding and simplifying
Vertex form
For example if an equation is 0=4(x-2)3+5 the vertex will be (2+3)
How did it get from -2 to 2 well that's because the x coordinate is always changed to a positive if the number is a negative or switched to a negative if the number is already a positive.
Standard form
Vertex form
Finding an equation when only vertex form is given
Graphing from vertex form
Start with the function in vertex form:
y = a(x - h)2 + k ( y = 3(x - 2)2 - 4)
Pull out the values for h and k.
If necessary, rewrite the function so you can clearly see the h and k values.
(h, k) is the vertex of the parabola.
Plot the vertex. y = 3(x - 2)2 + (-4)
h = 2; k = -4
Vertex: (2, -4)
Factored form
Factors and zeros
Rewrite the factors to show the repeating factor.
( x - 3)( x - 3)(2x + 5)
Write factors in polynomial notation. The a is the leading coefficient of the polynomial
P(x) = a( x - 3)( x - 3)(2x + 5)
Multiply ( x - 3)( x - 3)
Then multiply ( x 2 - 6x + 9)(2x + 5)
For right now let an = 1
The polynomial of lowest degree with real coefficients, and factors ( x - 3)2 (2x + 5) is: P(x) = 2x 3 - 7x 2 + 8x + 45.
The main concept is simply to multiply the factors in order to determine the polynomial of lowest degree with real coefficients.
Multiplying out of brackets and Quadratic expressions
Common factoring
Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved
Finding the GCF of a List of Integers or Terms
1)Prime factor the numbers.
2)Identify common prime factors.
3)Take the product of all common prime factors.
If there are no common prime factors, GCF is 1
Factoring simple trinomials
What i like to do for this type of factoring is just use mental math but others may find it difficult. To factor a simple trinomial you need to find 2 numbers that multiply to give you the C value but also add up to give you the B value
For example:
x^2 + 7x + 6
6 and 1 are factors of 6 that will equal 6 if multiplied and 7 if summed.
You will then use these 2 numbers part of the equation y=a(x-r)(x-s)
y=(x+6) (x+1)
Factoring complex trinomials
1. You first need to multiply the A value with the C value.
2. Next, find the common factor of the product you get and find 2 numbers that will add together to give you the B value
3. After you find the 2 numbers sub them into the equation and group them.
4. Find common factors of each group and you will then use the numbers inside the brackets as well as the 2 numbers you got from grouping.
Perfect Square
Factoring by grouping
Graphing factored form
A quadratic equation is a polynomial equation of degree of 2. The standard form of a quadratic equation is 0 = ax^2 + bx + c
where a, b, and c are all real numbers and a 0.
If we replace 0 with y , then we get a quadratic Function
y = ax^2 + bx + c
whose graph will be a parabola.
Standard form
Completing the square
Quadtratic Formula
Word problems
Revenue
Calculators are sold to students for 20 dollars each. Three hundred 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?
Let R represent Revenue
Let X represent increase or decrease
To solve this word problem we have to use a certain equation (Price)(People)
That gives us
R=(20+5x)(300-30x)
R=-150x² + 900 x + 600
Next we have to complete the square which gives us a Maximum Revenue of $7350 and a selling price of $35
Numeric
The sum of the squares of two consecutive integers is 365. What are the integers?
Let one number be x. Since the two numbers are consecutive, the other number is x + 1.
The sum of the squares is (x)^2 + (x + 1)^ 2 and it is equal to 365. The equation is (x) 2 + (x + 1)^2 = 365.
If you look at the question, it does not say maximum, minimum so we do not need the vertex. What we need to do, is to solve the equation (x)^2 + (x + 1)^2 = 365, by factoring or quadratic formula.
Expand and simplify the equation into standard form: 2x^2 + 2x – 364 = 0 Factor or use the quadratic formula to find the two solutions: x = -14, x = 13.
The two solutions to the equation will give us two possible sets of integers which are answers to the question: The first set of integers is -14 and -13, and the second set of integers is 13 and 14.