# Grade 10 Quadratics

## 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

Linear relations have first differences, whereas quadratic relations have second differences. Look at the table below for reference as to what this means.

## Distributive Property with Factored Form

You have probably learned about the distributive property from earlier grades and that is exactly what you need to know to expand and simplify factored form. A factored expression looks like this: (x+2)(x+3). The main thing you need to know is that when you expand and simplify an expression like this is that you need to multiply each number, variable and/or both by each number or variable in the second bracket. For more clarifications look at the picture below which shows how to do it step by step.

## Factoring Standard Form

Now that you have learned how to form standard form from factored form, you now need to learn how to factor standard form. You have been learning how to factor from earlier grades but only with two numbers. Now you are going to learn how factor with numbers and variables. Factoring is not as complicated as it sounds. It is explained step by step in the video below through the guess and trial method.
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## Completing the Square

Completing the square is the method we use to form vertex form from standard form. What you do is: divide 'b' by 2 and square root that number. Add that number into the equation but since we are not allowed to change the equation, subtract that number from the equation too. Now that you have that new number, it will act as your 'c'. Ignore the subtracted number and your original 'c'. Then factor the equation you have made which will be a perfect square trinomial. Then simplify the factored form. Now go back to the two numbers that you had put aside and collect the like terms. In the end it should look like a vertex form equation. For further clarifications look at the picture below.

## Solving Factored Form

Now that we have learned how to change from factored form to standard form and vice versa, it is now time to learn how to solve factored form equations. The method we use is similar to the method we used for solving linear equations. We are going to substitute 'y' as '0' but we are not going to isolate. This is where things change a little bit but you can use the picture below to clarify.

## Solving Standard Form

To solve standard form we do not use the same method we used for factored form. This one is a little more complicated. To solve a standard form equation we use the quadratic formula. It would be a good idea to memorize the formula. You can watch the video below to learn how to solve standard form through quadratic formula.

## Solving Vertex Form

To solve a vertex form equation we use the method of isolation. You need to substitute 'y' as '0' and then begin to isolate 'x'. This will give you your two x intercepts. To solve for the y intercept, you must substitute 'x' as '0'. If you need a more thorough explanation then you can look at the step by step instructions below.

## 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

## 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

Once you have learned how to solve factored form it is easy to graph. To graph factored form you would need to find the two x intercepts through solving by substituting 'y' as 0. Then you would need to find the axis of symmetry to find the vertex. To find the axis of symmetry you would need to add the two x intercepts and divide that number by two. The axis of symmetry will be the x of you vertex (x, y). Then to find the 'y' sub in the x as the axis of symmetry into the factored form equation. The answer you get will be the 'y' of the vertex. The vertex is the optimal value- the lowest or highest coordinate on the graph. Now you will have the vertex and the two x intercepts and you can easily graph the factored form equation.

## Graphing Standard Form

Most people find graphing standard form the most difficult because for this you need to use quadratic formula. To graph, you need to find the x intercepts through quadratic formula. Then you need to find the vertex through axis of symmetry. You cannot find the axis of symmetry the same way you find it in factored form. For this there is a special formula (in the picture). To find the axis of symmetry you need to divide negative 'b' by 2a. Then substitute the axis of symmetry into the standard form equation to get 'y'. The axis of symmetry and 'y' are the vertex (x, y). Now you have the x intercepts and the vertex and can easily graph this parabola.

## 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=-4.9t²+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?

## 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

Overall, I have enjoyed quadratics very much. I found this unit quite simple compared to other units we have already done. The few specific concepts I have enjoyed are: factoring, distributive property, quadratic formula. The reason I enjoy these concepts is because I like to do algebra and found these concepts easy. Furthermore, I like to use different methods and formula's to solve equations. Although I enjoyed this unit, there were a few things that I did not like which includes graphing and word problems. I did not like graphing because I do not like to apply my knowledge and put the information into a graph. I can easily graph, it is not difficult for me but it is not something I enjoyed as much. I also did not like the word problem concept because I found it somewhat difficult. The reason I found word problems difficult was because I did not know what formula or method to use to solve the word problem. I know how to do all the methods but I do not understand when to use them in a word problem. Once I understand what I need to use to solve a word problem the it becomes easy to solve. This is the main reason I do not enjoy word problems. This is why I want to improve on the word problem concept. I will improve by practicing and continue to solve word problems. By practicing I will be able to recognize when I need to use a certain method to solve. I have also done a lot of work that I am proud of in math class, specifically the quadratic unit. The best work I have done is the vertex form unit test (quadratics 1). I have done very well on this test because I was very comfortable with vertex form and I feel it is the most simple form of a quadratic equation. In the end, this was a fun and educational unit and I have learned a lot of information that I thoroughly understand and enjoy.

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

Everything we have learned starting from Quadratics 1 up to Quadratics 3 goes hand-in-hand.

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.