# Quadratic Relationships

### By: Dylan Seenarine

## What is a parabola ?

## Parabola In The World Today

## Components of Parabola

Optimal Value: The lowest or highest point the parabola can go. this is the y value in the vertex. The optimal value determines if the parabola goes upward or downward (max or min).

Vertex: The maximum or minimum point on the graph. The vertex consists of the axis of symmetry, which is the x coordinate and the optimal value, which represents the y value.

Y-intercept: The y-intercept is point on the parabola that hits y-axis

X-intercept: The x-intercept is the point on the parabola that hits the x-axis

Zeros and Roots: these are the x values and is when you set the y=o

## VERTEX FORM: Y=A(X-H)²+K

## QUADRATIC RELATIONS IN VERTEX FORM: Y=A(X-H)2 +K

• *(h, k)* is the vertex of the parabola, and *x = h* is the axis of symmetry.

• the *h* represents a horizontal shift (how far left, or right, the graph has shifted from *x* = 0).

• the *k* represents a vertical shift (how far up, or down, the graph has shifted from *y* = 0).

• notice that the *h* value is subtracted in this form, and that the *k* value is added.

If the equation is *y* = 2(*x* - 1)2 + 5, the value of *h* is 1, and *k* is 5.

If the equation is *y* = 3(*x* + 4)2 - 6, the value of *h* is -4, and *k* is -6.

## FINDING THE ZEROS IN VERTEX FORM

Step 1:Find the vertex. Since the equation is in vertex form, the vertex will be at the point (h, k).

Step 2:Find the y-intercept. To find the y-intercept let x = 0 and solve for y.

Step 3:Find the x-intercept(s). To find the x-intercept let y = 0 and solve for x. You can solve for x by using the square root principle or the quadratic formula (if you simplify the problem into the correct form).

Step 4:Graph the parabola using the points found in steps 1 – 3.

## Common Factoring

Common factoring should *always* be the first thing you look for when you have an equation in standard form. You should look to see if each of the terms in the equation has a similar component that each of the terms can easily be divided into. The common factor can either be a variable (x) or a coefficient (any number.)

Example 1: 8x + 6

Step #1: Find GCF ( what is the greatest # in common between 8 and 6)

GCF=2

Step #2: write solution with brackets

8x/2 + 6/2 = 4x + 3

= 2(4x + 3)

Example 2: 12x³ - 6x²

step #1: GCF= 6x²

Step #2: 6x²(2x-1)

check your answer!

6x²(2x-1)

=12x³ - 6x²

## FACTORED FORM: Y=A(X-R)(X-S)

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

## Expanding

example:

4(3+4x)

- multiply the number outside the bracket by everything inside the bracket

= 4 x 3 + 4 x 4x

= 12 + 16x

## Factoring Simple Trinomials

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.

Example:

2x2 – 3x – 35

2(-35) = -70

Factors of -70 that add up to give you -3 are -10 +7

Now sub those into the equation

2x2 – 10x +7x – 35

Find the GCF of each group you create

2x(x-5) + 7(x-5)

Now put them together and your answer is:

(2x+7)(x-5)

## Difference of Squares

When working with a difference of squares, you must check to see that

1. Each term is a squared term and

2. There is a *minus* sign between the two terms.

Two terms that are squared and separated by a subtraction sign like this:

a2 - b2

Useful because it can be factored into (a+b)(a−b)

A difference of squares is always a binomial. Both terms in the binomial have a square root but the second term is always a negative

Example:

9x^2-49

(3x-7) (3x+7) will give you the exact same answer

## Quadratic Formula

Using the quadratic formula is simple, all you need to do is input numbers from the standard form equation to the quadratic formula, and then solve the formula

## Perfect Squares

What a perfect square?

Perfect Square Trinomial is the product of two binomials. But, both the binomials are same. When factoring some quadratics which gives identical factors, that quadratics are Perfect Square Trinomial.

Example of a perfect square:

x² + 10x + 25

= (x + 5)(x + 5)

= (x + 5)²

## Factoring by grouping

## Standard form To Vertex form

Example:

2x^2 + 16x + 32

2(x^2 + 8x) + 16

2(x^2 + 8x + 16) - 32 + 16 (Divide b by two then square it (8/2^2))

2(x^2 + 8x + 16) -16 (Add in the opposite of the Squared number and multiply it by the coefficient in the front of the equation)

2(x + 4)^2 - 16

## Word Problems

**Example.** One number is the square of another. Their sum is 132. Find the numbers.

Let A and B be the numbers. The first sentence says one is the square of the other, so I can write

The sum is 132, so

Plug into and solve for B:

The possible solutions are and .

If , then .

So two pairs work: -12 and 144, and 11 and 121

**Example.** The difference of two numbers is 2 and their product is 224. Find the numbers.

Let x and y be the numbers. Their difference is 2, so I can write

Their product is 224, so

From , I get . Plug this into and solve for y:

If , then .

If , then .

So two pairs work: -14 and -16, and 14 and 16.