# Quadratics

### 101

## Table of Contents

2. Parabolas

3. Vertex form

4. Graphing from factored form

5. Expanding

6. Multiplying Binomials

7. Common Factors

8. Factoring Simple Trinomials

9. Factoring Complex Trinomials

10. Perfect Squares

11. Differences of Squares

12. Quadratic formula

13. Discriminant

14. Standard to Vertex Form

15. Reflection on Unit

## First & Second Differences

Second differences are used to tell if a relation is quadratic or not. In a quadratic relation the first differences don't have to be the same, but the second differences must.

## Parabolas

A parabola is a symmetrical curve (both sides of the parabola are the same)

Properties of Parabolas:

Vertex- The vertex is the lowest point of the parabola and is located in the center parabola

Axis of Symmetry- A vertical line that divides the parabola in half perfectly

Y intercept- The point where the parabola crosses through the Y axis

X Intercept- The point where the parabola crosses through the X axis

Min/Max Value- The Y value of the vertex

A quadratic relationship can be graphed by using a chart to organize the x and y as shown above

## Vertex Form y=a(x-h)^2+k

- Axis of symmetry

-Optimal Value

-X-intercepts or zeros

-Transformations

-Step pattern

## Axis of Symmetry

## Optimal Value

## X Intercepts/Zeroes

y=3(x+3)^2-12

0=3(x+3)^2-12 Sub Y in as 0

12=3(x+3)^2 Take 12 to the other side of the equation

12/3=(x+3)^2 Divide both sides by 3

4=(x+6)^2 Square booth sides by - & +

-+2=x+6 Now calculate for the zeroes

-2=x+6

-2-6=x

-8=x

2=x+6

2-6=x

-4=x

## Transformations

- a controls if the parabola opens up or down and how stretched/ compressed the parabola is
- h controls the horizontal shift
- k controls the vertical shift

If the A value is positive then the parabola will open upwards, if it is negative then it will open downwards.

If the A value is greater than one then the parabola will be stretched. If it is smaller than one but greater than zero it means the parabola is compressed.

The H value represents the horizontal position of the vertex in relation to the origin. If the h value is negative then the vertex shifts to the right, if it is positive the vertex shifts to the left.

The K value represents the vertex of the parabola on the Y axis.

## Step Patterns

## Graphing From Vertex Form

## Factored Form y=(x-s)(x-r)

## Axis of Symmetry

y=(x-4)(x-8)

4+8=12

12/2=6

The axis of symmetry for this equation would be 6.

## Optimal Value

y=(x-4)(x-8)

y=(6-4)(6-8)

y=(2)(-2)

y=-4

The optimal value would therefore equal -4 in this example.

## Finding X intercepts/zeroes

## Graphing from Factored Form

- Find the axis of symmetry
- Sub it into the equation to find the y value of the vertex
- Plot the vertex & zeroes then connect them

## Expanding

Example:

3(4+3x)

=(3*4)+(3*3x)

=12+9x

## Multiplying Binomials

Example:

(x-4)(x+5)

=x^2+5x-4x-20

=x^2+x-20

## Standard Form y=ax^2+bx+c

## Common Factoring

## Simple Factoring

x^2+6x+8

(x+4)(x+2)

To properly factor you must find two numbers that add up to the B but when multiplied equal the C. 4 and 2 when added equal 6 and when multiplied equal 8.

## Factoring Complex Trinomials

*Example:*

Factor the following trinomial.

*2x^2* +10*x*+12

*Solution*:

Step 1:The first term is 2x^2 *, *which is the product of 2*x* and *x*. Therefore, the first term in each bracket must be *x*.

*2x^2+10x+12*

Step 2: First you want to simplify the equation by common factoring

2x+10x+12

=2(5x+6)

Step 3: Next you should look for factors of the two numbers in the brackets in this case those numbers are two and three.

2(5x+6)

=2(x+2)(x+3)

Therefore the factored form of 2x^2+10x+12 is 2(x+2)(x+3)

## Perfect Squares

x^2+12x+36

(x+6)^2

This example is a perfect square because 6+6 equals 12(b) and 6*6 equals 36(C)

Another example:

x+6x+9

(x+3)^2

3+3 equals 6(B) and 3*3 equals 9(C) which makes this equation a perfect square.

## Differences of Squares

Example:

x^2-25 (There is no b because one factor is negative while the other is positive and they are the same number so they cancel each other out.)

Two factors must be the same and must still multiply to equal C.

x^2-25=(x+5)(x-5)

The simple way to find these factors is to square root your C and change the operations of that number.

Example:

x^2- 25

x^2 + 5^2

(x+5)(x-5)

Example:

3x^2+6x+2

If you take this equation and put it into standard form it would be:

-6+-√36-4(3)2

--------------------

2(3)

Solution:

-6+-√12

-6+3.46=-0.42

------------

6

-6-3.46=-1.57

----------

6

The zeroes of this equation would be 1.57 and 0.42

## Discriminants

If D is greater than 0 it means there are two zeroes

If D is less than 0 it means there are no zeroes

If D is equal to 0 it means there is only one zero.

## Links Between Topics

## Word Problems

ANSWER: 12 and 9

The equation shows the height (h) of a baseball in metres as a function of time (t) in seconds.

h=−5t2+40t+3

a) What was the initial height of the ball?

b) At what time does the ball hit the ground?

ANSWERS:

A)3M

B)8.07 Seconds