# Grade 10 Quadratics

### BY: Karman Gakhal

## Table of contents

Introduction picture

1. Expanding and Simplifying

- Factoring Standard Form
- Common Factoring
- Factoring by Grouping
- Simple Trinomials
- Complex Trinomials
- Difference of Squares
- Perfect Squares
- Completing the Square

2. Graphing

-Standard Form

- x Intercepts
- Axis of Symmetry

- Each Part of Equation
- Types of Graphs
- Transformations
- How to Graphs

- x Intercepts
- Axis of Symmetry
- Optimal Value

3. Standard form

- Learning Goals
- Summary
- Quadratic Formula
- Completing the Square
- Word Problem

4. Solving

- Solving Standard Form
- Solving Vertex Form

- Solving Factored Form

5. Word Problems

6. Reflection

7. Connections

8. Useful links

## What is a parabola?

## Solving: Expand and Simplify

## Factoring Standard Form

## Learning Goals

how to factor by grouping

How to factor simple trinomials

How to factor complex trinomials

How to find the difference of squares

How to find factor perfect squares

Basic formula for factoring

## Summary

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

The value of r and s give you the x-intercepts

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

## Common Factoring

## Video

## Factoring by Grouping

## Simple Trinomials

## Video

## Complexed Trinomials

## Video

## Deference of Squares

## Video

## Video

## Completing the Square

## Word Problem

1)Write the formula in factored form

h=-2(t-6)(t+4)

2)When will the ball hit the ground

6 seconds (t=6)

3)What is the maximum height the ball will reach in metres?

50m

## Graphing

## Standard Form

## Vertex Form

## Learning Goals

2. How to graph vertex form

3. Equation for vertex form

4. Variables in vertex form

5. What a standard graph looks like

6. How to make a standard graph match your parabola

7. First and Second difference

8. Step Pattern Method

## Each Part of Equation

H= tells us how many units left/right the vertex is going to move from 0

K= tells us how many units up/down the vertex is going to move from 0

X= the the 'y' are coordinates on the graph

## Transformations

Math has its own language and there is specific terminology that you must know. When explaining the transformations of certain coordinates of a vertex form graph, there are certain words we use. These words are:

- Vertical Stretch: If the 'a' is a number greater than 1 then the graph would have a vertical stretch which means that is steeper. The way you would write it: This graph has been vertically stretched by the factor of 2 (or any number greater than 1).
- Vertical Compression: If the 'a' is a number less than 1 then the graph would have a vertical compression meaning it would be wider. The way you would write it: This graph is vertically compressed by the factor of 0.5 (or any number less that 1).
- Vertical Reflection: If the 'a' is a negative number then it means the graph is vertically reflected over the x axis downwards.
- Translations: If there is a number after x squared like in y=a(x-h)²+5 (the 'h') then it means that the vertex has been translated right or left. The way you would write it: The vertex has been horizontally been translated 3 (or any other number) units to the right/left. Also if there is a 'k' then that means that the vertex has been translated up or down. The way you would write it: The graph has been vertically translated 4 (or any other number) units up/down.

## Step patterns (Basic)

Over 1 -- Up 1

Over 2 -- Up 4

## Types of Graphs

## maximum This is an example of a maximum graph because it is opening downwards. | ## minimum Unlike the maximum this graph is opening upwards causing it to be a minimum graph. | ## 2 X-intercepts This graph has 2 X-intercepts or otherwise known as zeros because 2 points of the parabola are touching the x axis. |

## 1 x-intercept This parabola only has one point of it touching the x intercept in cases like this the vertex is always touching the x axis. | ## no x-intercept Like the name suggests this parabola has no points touching the x axis. |

## How to Graphs

## solving

## Solving Standard Form

## Quadratic Formula To solve a standard form equation you need to use the quadratic formula which is shown in the picture. Then substitute the variable with the numbers from the equation to find the x intercepts. | ## Discriminant If the number in the square root is a positive number like shown in the example then you will have 2 real solutions. If the number in the square root is a negative number then you will have no real solution since you cannot square root a negative number. Finally, if the number in the square root is 0 then you will have only one solution. | ## X-Intercepts Make sure you remember the fact that when you square root something, the outcome can be both positive and negative. This is why you need to find x both ways and will end with 2 'x' intercepts. |

## Quadratic Formula

## Discriminant

## Solving Vertex Form

## Y-Intercept To find the y-intercept, you need to sub 'x' as '0'. After subbing 'x' as '0', you can solve for the y-intercept. | ## X-Intercept part 1 To find the x-intercept, you need to sub 'y' as '0'. After subbing 'y' as '0', you can solve it. | ## X-Intercept part 2 You want to isolate 'x' as its own, so you need to square root 6, and whatever you do on one side, you need to do it on the other side. Then you continue solving to receive 2 x-intercepts. |

## Y-Intercept

## X-Intercept part 1

## Factored Form

## Factored Form This is how a factored form equation looks like and how you would solve to find your zeros. | ## Zeros & Axis of Symmetry After finding the zeros, you need to find the AOS, which is done by adding the two zeros and dividing it by '2'. | ## Optimal Value To find what 'y' is equal to, you need to sub in x=-0.5 (this is the AOS/x-intercept). |

## Factored Form

## Zeros & Axis of Symmetry

## Standard Form

## Learning Goals

- What is the quadratic formula used for
- what is the quadratic formula
- how to complete squares

## Summary

## Quadratic Formula

## Completing the Square

## Word Problems

## flight

A ball is thrown upwards at an initial velocity of 8.4m/s, from a height of 1.2 m above the ground. The height of the ball, in meters, above the ground after t seconds is modeled by the equation *h=-4.9t*²*+8.4t +1.2.*

1. How long will it take for the ball to fall to the ground, rounded to the nearest tenth of a second?

2.What is the maximum height of the ball? At what time will it reach this height? Round your answers to the nearest tenth.

## Factor Quadratic Expressions

The flight of a ball is modeled by the equation ℎ = −5² + 20 + 25 where h represents the height of the ball in meters, and t represents the amount of time the ball has spent in the air in seconds.

a. Write the equation in factored form, determine the x-intercepts and vertex, and graph.

b. When does the ball hit the ground?

c. What is the highest the ball flies, and when does that happen?

## Reflection

## connections

In quadratics 1, we mostly focused on solving vertex form by finding axis of symmetry, optimal value, zeros, and determining the step pattern. With all this information, we were able to graph it.

In quadratics 2, we focused on expanding and factoring. We learnt how to find axis of symmetry, optimal value, and zeros. We also did this in quadratics 1, but the only difference was that there might have been some different methods to do this.

In quadratics 3, we focused on rewriting standard form equations into vertex form by completing the square and by learning a new equation which is called the Quadratic Formula. We also learnt how to graph quadratics using the x-intercepts which we also learnt in quadratics 1.

In quadratics 1, 2 & 3, the word problems were mostly all asking for the same stuff just in different forms of equations and different methods of asking them. For example, what is the max height? They are asking for you to find the vertex and state the 'y'. In vertex form {y=a(x-h)²+k}, the vertex is (h,k) and so you already know your max height by just looking at the equation and knowing 'k' is your 'y-intercept'. In standard form {y=ax²+bx+c}, you know that the 'c' is the 'y-intercept' which is your max height. Now in factored form, you have to do some solving to find the max height, starting with finding the zeros, then the AOS, & lastly the optimal value. The AOS and optimal value would be your vertex, at which the optimal value is the 'y-intercept'.

As you can see, all of the parts of quadratics are related in several ways.