# Grade 10 Quadratics

### Graphing, Factoring, Solving

## By: Simran Gakhal

## Table of Contents

***Quick Look Back to Quadratic Relations**

1. Expanding and Simplifying

- Distributive Property with Factored Form
- Factoring Standard Form

-Simple Trinomials

-Complex Trinomials

-Difference of Squares

-Perfect Squares

-Common Factoring

- Completing the Square

2. Solving

- Solving Factored Form
- Solving Standard Form
- Solving Vertex Form

3. Graphing

- Vertex Form

- Vertex and Step Pattern

- Transformations

- Factored Form

- x Intercepts

- Axis of Symmetry

-Optimal Value

- Standard Form

- x Intercepts

- Axis of Symmetry

4. Word Problems

5. Reflection

6. Connections

## Linear & Quadratic Relations

## EXPANDING AND SIMPLIFYING

## Distributive Property with Factored Form

## Factored Form This is how a quadratic relation looks when it is factored. | ## Distributing You have to multiply the x in the first bracket by the x and the 3 in the second bracket. | ## Distributing Then, you multiply the 2 in the first bracket by the terms in the second bracket. |

## Multiplying Once you have distributed all of the terms you should have x²+3x+2x+6 | ## Adding Like Terms Now that there are no more brackets and no more multiplying to do, you can add the like terms. This means add the like terms which are 3x and 2x in this case. | ## Final Product Your final product should have the expression you started off with in factored form now in standard form. Expanding and simplifying factored expressions always end in standard form equations. |

## Adding Like Terms

## Factoring Standard Form

## Solving Complex Trinomials

## Solving Perfect Squares

## Solving Difference of Squares

## Completing the Square

## SOLVING

## Solving 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

## 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

## Vertex Form This is how a vertex form equations looks like. | ## 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 To find the x-intercept, you need to sub 'y' as '0'. After subbing 'y' as '0', you can solve |

## X-Intercept 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. |

## GRAPHING

## Graphing Vertex Form

It is very easy to graph a vertex form equation because most all of our information is seen in the equation. A vertex form equation looks like y = a(x-h)²+k

- The 'a' tells us the stretch/ compression of the graph
- The 'h' tells us how many units left/right the vertex is going to move from 0
- The 'k' tells us how many units up/down the vertex is going to move from 0
- The 'x' and the 'y' are coordinates on the graph

One key point you need to know is that (h, k) are the vertex. This is why it is easier to graph a vertex form equation because the vertex is given in the equation.

You also need to know that when your parabola is going to open upwards or downwards. This information is also found in the equation. If the 'a' is negative then your graph will open downwards and if the 'a' is positive, it will open upwards.

**STEP PATTERN**

-2 ........... -4

-1 ........... -1

*0 .............. 0*

*1 .............. 1*

*2 .............. 4*

**Basic Pattern:**

Over 1 -- Up 1

Over 2 -- Up 4

## Graphing Vertex Form It is easy to graph from vertex form because the 'h' and the 'k' are the vertex (h, k)/ (x, y). Also, you use the step pattern to find the other coordinates of the graph. In this example, the 'a' is two. This means that every time you move one unit right from the vertex, you move 2 units up to find the next coordinate, then if you move 2 units to the right, you move 4 units up to find the next coordinate and so on. | ## Positive If the 'a' in a vertex form equation is a positive number then it means that the graph will open upwards. | ## Negative If the 'a' in a vertex form equation is a negative number then it means that the graph will open downwards. |

## Graphing Vertex Form

## Positive

## 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.

## Graphing Factored Form

## Graphing Standard Form

## Flight

To solve a flight word problem you need to know the quadratic formula and the axis of symmetry. Here is an example of a flight word problem:

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 of the Form ax² + bx + c

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?

## X-Intercepts & Vertex For part a., you need to find the x-intercepts and the vertex which is explained in the picture. | ## Graph After you find the x-intercepts and the vertex, you can easily graph it. | ## Using the X-Intercepts and Vertex to answer other questions For part b., they are asking when the ball hit the ground, which means they are asking for the x-intercept. For part c., they are asking when the ball hits its highest point and when it happens which means they are asking for the vertex. |

## X-Intercepts & Vertex

## Optimization

You have a 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area?

## Reflection

## Multiple Choice I can quite easily eliminate different answers from multiple choice. | ## Short Answer I easily understand the question and know what I have to do to get my answer. | ## Factored Form & Vertex Form Question I can easily gather the information that the question is asking me. Using that information, I can graph. |

## Application 1 I know what the question was asking me for each part & what I had to look for. | ## Application 2 Once again, in this question, I knew what I was looking for which made it easier for me to know what to do. | ## Communication In this communication question, I was asked to do some extra thinking and how I can figure out if that equation had any x-intercepts or not. At first, I remember having a blank face at this, not knowing what to do. But then, I thought about it and read the question several times. I realized that I already knew the answer because all my information is right there. So all I had to do was to just write down all that information I was given and explain how it could be used. |

## Application 2

## Communication

## How does Quadratics 1, 2 & 3 relate?

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.