# Quadratic Relationships

### Sarah Qarizadha

## Welcome Everyone!

## Learning Goals

- To be able to solve all vertex form equations
- Finding the value of
*'a'* - To use quadratic equations to solve word problems in vertex form

## Introduction

## Definitions Of Unit #1

__Parabola__ : This is a curve, shaped as a arch. Its distance from a fixed point is equal to its distance from a fixed line

__Vertex Form__ : The vertex are the points (*h,k*) on a parabola. The vertex is when both the points meet. The *x*-intercept is *'k'* and the* y*-intercept is *'h'*.

__AOS (Axis Of Symmetry)__ : The AOS is a straight vertical line that goes right down the middle of the parabola

## Introduction To Vertex form

- Vertex is (
*h,k*) *'a'*tells us if its stretched or compressed, and the direction of opening*'h'*tells us if its going to be left or right on the graph/horizontal translation*'k'*tells is if its a vertical translation- Use the step pattern
- To find the y-intercept, set x=0 and then solve for the y-intercept
- Then to solve, set y=0 and solve for x or expand and simplify to get the standard form
- Then use the Quadratic Formula

## Equation Question Examples

__Example #1 - Determine an equation when given the vertex and determining the 'a' value:__

Vertex : (*-3,5*)

One of the points are (*3,7*)

__Step 1__ : Plug in the vertex into the equation

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

y=a(x+3)^2+5

__Step 2__ : Find the *'a'* value. Plug in (*3,7*) in the equation

y=a(x+3)^2+5

7=a(3+3)^2+5

7=a(36)+5

7=36a+5

7-5=36a

2/36 = 36a/36

18=a

__Step 3__ : Fill in final equation

y=18(x+3)^2+5

__Example #2 - Determining an equation when given the vertex and when giv____en (x,y) __:

__Example #3 - Isolating For 'x' :__

*We are finding the *x*-intercept when we are isolating for *'x'*

__Graphing Vertex Form__

We can easily use a step pattern to graph y = 2(x-4)^2 - 6

In the step pattern, your supposed to multiply the *'a'* value by 1, 3 and 5

Vertex : (4,-6)

AOS : -4

Optimal value : -6

*'a'* : 2

Direction Of Opening : Up, because the *'a'* value is positive

By using the step pattern, we can easily figure out what parabola will be :

- Step 1 : 1x2 = 2
- Step 2 : 3x2 = 6
- Step 3 : 5x2 =10

## Word Problem

## Using Vertex Form

**At a baseball game, a fan throws a baseball from the stadium back onto the field. The height in meters of a ball t seconds after being thrown is modeled by the function h = -4.9 (t-2)^2 + 45**

**a) What is the maximum height of the ball?**

The Vertex : (*2,45*)

Maximum height of the ball is 45 meters

**b) When did the maximum height occur?**

The maximum height is 2 seconds

**c) What is the height of the ball after 1 second?**

*h* = -4.9 (1-2)^2 + 45

*h* = 40.1 meters

Therefore, the height of the ball is 40.1 meters after 1 second

**d) What is the initial height of the ball?**

When* t*=0 Solve for *h*

*h* = -4.9 (0-2)^2 + 45

=25.4

Therefore, the initial height was 25.4 meters

## First and Second Differences

## Mapping notation

Example - y=(x+7)^2 ~ (x-7, y)

## Graphing Using Transformations

## Factored Form

## Learning Goals

- To be able to graph a quadratic equation in factored form
- Turning Factored Form, Quadratic Equations into Standard form by using FOIL and simplifying

__Factors And Zeros__

A zero of a parabola is another name for the x- intercepts and in order to find the x- intercepts you must set y=0

Key points

- When the (a) value changes the zeroes do not change
- When the (a) value changes the axis of symmetry do not change
- When the (a) value changes the optimal value does change

## Introduction To Factored Form

*a*gives you the shape and direction of opening

The value of

*r*and

*s*give you the

*x-*intercepts

axis of symmetry, AOS: =

*(r+s / 2)*

- Sub this
*x*value into equation to find - the optimal value
- to find the
*y*-intercept, set*x*=0 and solve for*y* - Solve using the factors

Types of Factoring:

- Greatest Common Factor
- Simple factoring (
*a*=1) - Complex factoring
- Special case - Difference of squares
- Special case – Perfect square

## Equation Question Examples

## Factoring Complex Trinomials

## Factoring by grouping

Step 1: Decide if the four terms have a GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 2: Group first two terms together and the last two terms together.

Step 3: Factor out the GCF from each of the two groups.

Step 4: The one thing that the two groups have in common should be what is in parenthesis, write whats outside the brackets as a parenthesis

Step 5: Determine if the remaining factors can be factored any further.

## Graphing Factored Form

## Common Factoring

Step 1: Determine the GCF of the given terms. The greatest common factor or GCF is the largest factor that all terms have in common.

Step 2: Factor out or divide out the GCF from each term. You could check your answer at the point by distributing the GCF to see if you get the original question. Factoring out the GCF is the first step in many factoring problems.

## Word Problem Using Factored Form

*h*

*= -5*

*t*^2 + 25

*t*where

*h(t)*is the height in meters and

*t*is the time in seconds. When will the rocket hit the ground?

h = -5*t* ^2 + 25*t*

*h* = -5*t* (*t*-5)

*t*=0 *t*=5

Therefore, the rocket hit the ground at 5 seconds

## Special Cases

## Difference Of Squares

__Equation__

a^2 - b^2 = (a - b)(a + b)

or

a^2 - b^2 = (a + b)(a - b)

Step 1: Decide if theirs a GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 2: So, all you need to do to factor these types of problems is to determine what numbers squares will produce the desired results.

Step 3: Determine if the remaining factors can be factored any further.

## Perfect Squares

__Equation__

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

Step 1. Verify that the first term and the third term are both perfect squares. (This means that the coefficients are perfect squares

Step 2. Verify that the middle term is twice the product of the square roots of the first and third term.

Step. 3. Use the standard form above to write the factored form.

## Factored Form - MY VIDEO

## Standard Form

## learning goals

- Find the number of zeros that a quadratic relationship has by calculating the discriminant.
- Solve using the quadratic formula or by factoring or by completing the square to get vertex form

## Introduction to standard form

The value of *a *gives you the shape and direction of opening

The value of *c *is the *y*-intercept

Solve using the quadratic formula, to get the *x*-intercepts

MAX or MIN? Complete the square to get vertex form

## Quadratic FORMULA example

**: A discriminant is to know how many x-intercepts there will be. Using this formula: b^2 - 4ac**

__The Discriminant__- There can be two solutions, one solutions or no solution.
- There are two solutions when the discriminant is a positive.
- There is one solution when the discriminant equals to 0
- There is no solution, when the discriminant is negative

**:**

__Turning Into Vertex Form And Completing The Square____Question__: y=3x^2 +6x -4

__Step One__: Focus on the 3x^2 +6x, and factor out the GCF (which is 3 in this case) and leave the -4 out.

y=3(x^2 +2x) -4

__Step Two__: Divide the 'b' value by 2 and then square the result.

__Step Three__: Add the result in the bracket and also subtract it. So the equation looks like this: y= 3(x^2 +2x +__ -__) -4

__Step Four__: Take the negative one outside of the brackets, therefore it will be multiply with 3.

__Step Five__: factor the trinomial in the bracket.

- *Remember Vertex Form Equation* y=a(x-h)^2+k
- Therefore, vertex is (-1, -7)

## word problem using quadratic formula

*t*seconds is modeled by the equation

*h= -4.9t^2 + 51t +1.3*

a) How long does it take the rocket to fall to the ground, rounded to the nearest hundredth of a second?

b) Find the times when the toy rocket is at a height of 95.7 m above the ground. Round your answers to the nearest tenth.

c) What is the maximum height of the toy rocket? At what times does it teach this height? Round you answer to the nearest tenth.

## Video explaining standard form - Quadratics

## REFLection on quadratics

## Connections between quadratic relations

*h*and

*k*). Another connection i made between units, was the graphing. when we graph for any unit, you can always find the axis of symmetry (AOS). By graphing, you can also find the x-intercepts/zeros. By completing either Vertex, Standard or Factored form, you can find the vertex for any of the equations and then graph it, which is also a similar connection between all the units. These are some connections you could make between all three units.