# QUADRATIC RELATIONSHIPS

### Have Math in your Path!

## TABLE OF CONTENTS

- What is Quadratics
- What is a Parabola?
Parabolas in Real Life

- Introduction to Quadratics
- Quadratic Relations
- Investigating Vertex Form
- Factored Form
- Multiplying Binomials
- Common Factors and Simple Grouping
- Simple Trinomials
- Complex Trinomials
- Perfect Squares

- Difference of squares
- Recap
- Completing the square
- Solving quadratic relations
- Standard form in algebra tiles
- Quadratic formula
- Recap
- Connections
- Reflection

## WHAT IS QUADRATICS?

*x*raised to the power of two.

## WHAT IS A PARABOLA?

A symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity ideally follows a curve of this shape. You will know that it is a quadratic equation if it has x².

## PARABOLAS IN REAL LIFE

## Introduction to Quadratics

**KEY FACTORS OF QUADRATIC RELATIONS**

The graph of a quadratic relation is called a parabola. The Parabola has some important features......

- Vertex: The point where the axis of symmetry and the parabola meet, it is the point where the parabola is at its maximum value (if the direction of opening is down) or minimum value (if the direction of opening is up) and it is the point where the graph changes directions. It is labelled as (x,y).
- Optimal Value/ Minimum or Maximum Value: The value of the y co-ordinate of the vertex. It can be the lowest or highest values/ points. It is labelled as "y".
- Axis of Symmetry: A vertical line that goes through the vertex and divides the parabola into two equal halves. It is labelled as "x".
- Y- Intercept: Where the graph crosses the y-axis. It can be labelled as (0,y).
- X- Intercept: Where the graph crosses the x-axis. It can also be called as "zeros" or "roots". It can be labelled as (x,0).

**Big Idea – need more than a line, some things go up and then down, they have x- intercepts, a vertex, axis of symmetry and an optimal value**

## Quadratic Relations

**USING TABLE OF VALUES, TECHNOLOGY AND FIRST DIFFERENCES TO ANALYZE QUADRATICS**

You can determine if an equation is linear or quadratic by the use of a table of values.....

- If the relationship is linear, the first differences on the chart are constant
- If the relationship is quadratic, the second differences on the chart are constant, but the first differences are not

You can also determine if an equation is linear or quadratic by plotting the points on a graph...

- If the relationship is linear, the points on the graph lie along a line

- If the relationship is quadratic, the points on the graph will not lie along a line, but will form a curve

- is upright
- has a minimum turning point at (#, #)
- has an axis of symmetry : x = #
- has y-intercept at (0, #)
- has x-intercept at (#, 0)

**WORD PROBLEM **

1. A baseball is tossed into the air and follows the path h = -2t²+ 6t , where t is the time, in seconds, and h is the height of the baseball, in metres.

a) Sketch the path of the baseball.

This question is asking for the optimal value (also known as the y value of the vertex). By analyzing the graph, you come to the conclusion that the maximum height of the baseball is 4.5m.

c) At what time will the baseball reach its maximum height?

This question is asking for the value of then x value in the vertex. By analyzing the graph, you come to the conclusion that the baseball reaches its maximum height of 4.5m at 1.5 seconds.

## Investigating Vertex Form

The standard form of the equation of a quadratic function: y= ax² + bx + c can go through a variety of transformations to form y= a (x-h)² + k.

1. To graph y= ax² ,stretch or compress the graph of y = x² vertically by a factor of "a"

- if a < 0 (so if it's negative), the parabola is reflected in the x-axis
- If a > 1 or a < -1, then the graph is stretched vertically (narrows)
- If -1 < a < 0 or 0 < a < 1, then the graph is compressed vertically (widens)

2. To graph y= x²+ k, translate the graph of y = x² vertically k units

- if k > 0 (so if it's positive), then the graph is translated k units upward
- if k < 0 (so if it's negative), then the graph is translated k units downward

3. To graph y= (x- h)², translate the graph of y = x² horizontally h units

- if h > 0 (so if it's negative), then the graph is translated h units to the right
- if h < 0 (so if it's positive), then the graph is translated h units to the left

**VERTEX FORM WITH STEP PATTERN**** **

The step pattern is a pattern that identifies your next points in each parabola.

The original step pattern is "Over 1 up 1 and Over 2 up 4"

The value of how many steps up is the square of the value of the steps over.

This pattern shows you that when the point moves 1 over from the vertex, it then moves 1 up, and this pattern continues.

But in quadratic relations if there is an "a" value the step pattern changes.

If there is an "a" value then the squared number in the pattern needs to be multiplied by the "a" value.

Example: y= -2 (x-3)² + 5

Vertex: (3,5)

The step pattern would now be "Over 1 up -2 and Over 2 up -8"

**WORD PROBLEM INVOLVING VERTEX FORM**** **

1. The area available for a swimming pool in a backyard is a square with side length 20m. The square swimming pool is to be placed in the centre of the area. If the side length, in metres, of the swimming pool is x, then the area of the backyard remaining is given by the relation A = -x² +400.

a) Graph the relation.

b) What are the intercepts? What do they represent?

By analyzing the graph, you come to the conclusion that the y- intercept is 400. This represents the area of the backyard if there is no swimming pool. You also come to the conclusion that the x-intercept is 20. This represents the side length of the swimming pool, in metres, if the pool completely fills the backyard.

**FINDING EQUATIONS AND ANALYSING VERTEX FORM**

1. Sub the coordinates from the vertex into the equation.

- remember: y= a (x-h)² +k
- vertex= (h,k)
- h= opposite value (positive= negative and negative= positive)

- sub the y value into the y of the equation
- sub the x value into the x of the equation

4. Write the equation.

**Example: Analysing the given information from an equation to graph **

Example 1: Write an equation for a parabola with vertex (3, -4) and passing through (2, -7).

y = a(x - h)² + k

The question provides the vertex, (3, -4). Thus (h, k) = (3, -4).

y = a(x - 3)² - 4

(2, -7) is on the quadratic. Thus (x, y) = (2, -7). Substitute these values into the equation:

**HOW TO COLLECT INFORMATION THAT CAN BE USED FOR GRAPHING IN VERTEX FORM**** **

By using the vertex form, you can easily gather the information needed to graph the parabola (ex. you will find the vertex, direction of opening, axis of symmetry, etc.)

__Example 1: y= (x+3)²__

*Vertex: *(-3,0)

*Axis of symmetry: *X=-3

*Stretch or compression factor relative to y=x**²: *None

*Direction of opening: *Upward

*Value of x: *X=-3

*Value of y: *Y=0

__Example 2: y=(x+2)²+5__

*Vertex: *(-2,5)

*Axis of symmetry: *X=-2

*Stretch or compression factor relative to y=x**²: *None

*Direction of opening: *Upward

*Value of x: *X= -2

*Value of y: *Y=5

__Example 3: y=3(x+7)²-2__

*Vertex: *(-7,-2)

*Axis of symmetry: *X=-7

*Stretch or compression factor relative to y=x**²: 3*

*Direction of opening: *Upward

*Value of x: *X= -7

*Value of y: *Y=-2

**REAL LIFE SITUATION INVOLVING VERTEX FORM #1**

The path of a football is modelled by the relation h= -0.25 (d-12)² +36, where d is the horizontal distance, in metres, after it was kicked, and h is the height, in metres, above the ground.

a) Sketch the path of the football.

The question is asking for the maximum height, which means that you need to find the optimal value (also known as the y-value of the vertex). In the form y= a (x-h) ² +k, *k *tells you the optimal value, which is the maximum height. Therefore, the maximum height of the football is 36m.

c) What is the horizontal distance when this occurs?

The question is asking for the horizontal distance when the football is at it's maximum height, which means that you need to find the x-value of the vertex. In the form

y= a (x-h) ² +k, *h *gives you the x-value of the vertex, but it needs to be changed to the opposite sign. Therefore the horizontal distance when the football is at it's maximum height is 12m.

d) What is the height of the football at a horizontal distance of 10m?

Since *d* represents the horizontal distance, we need to set the value of *d *in the equation h= -0.25 (d-12)² +36 to 10.

h= -0.25 (10-12)² +36

h= -0.25 (-2)² +36

h= -0.25 (4) +36

h= -1+36

h= 35

Therefore, the height of the football at a horizontal distance of 10m is 35m.

**HOW TO FIND THE Y-INTERCEPT IN VERTEX FORM?**

Just to remind, the equation of a vertex form is y= a(x-h)² + k

To find the y-intercept, first substitute y = 0.

Then solve the equation step by step to find the y-value.

**HOW TO FIND THE X-INTERCEPTS (ZEROS) IN VERTEX FORM?**

Here are the step-by-step instructions:

- Set y = 0
- Move k to the other side.
- Divide both sides by "a"
- Square root both sides
- Move h to the other side.
- You will get solutions, solve each of them

**REAL LIFE SITUATION INVOLVING VERTEX FORM #2**

A football is punted into the air. Its height h, in metres, after t seconds is

h = -4.9(t - 2.4)² + 29

a) What was the height of the ball when it was kicked?

b) What was the maximum height of the ball?

*The maximum height is the optimal value of the parabola.*

*The k value in the equation represents the optimal value.*

Therefore, the maximum height of the ball is 29 m.

c) When does the ball hit the ground?

*Here we have to find the value of t when h = 0*

*Because we are looking for the time in seconds when the height is 0.*

**VIDEO: GRAPHING USING VERTEX FORM**

**Big Idea – Vertex Form tells us about what the graph looks like**

## Factored Form

- Factored Form Equation: y= a (x-r) (x-s)
- Zeros: x-intercepts= (r,0) (s,0) the r and s value are opposite values (positive= negative and negative= positive)
- Axis of Symmetry: a vertical line between the two zeros; gives you the x co-ordinate of the vertex
- formula to find the x co-ordinate of the vertex: x= r+s / 2

- optimal value: sub the x co-ordinate of the vertex into the equation to find the optimal value (the y co-ordinate of the vertex)
- direction of opening: up= a value is positive and down= a value is negative

**WHAT EACH VARIABLE REPRESENTS**

-The zeros/x-intercepts: are the r and s value you find them by setting each (factor) equal to zero.

-The axis of symmetry: is the midpoint of the two zeroes

-The optimal value: is found by subbing the axis of symmetry value into the equation

Example 1: y=0.5(x+3)(x-9)

1) find the zeroes

y=0.5(x+3)(x-9)

0=0.5(x+3)(x-9)

x+3=0 and X-9=0

x=0-3 and x=0+9

x=-3 and x=9

2) find the axis of symmetry

X=-3+9/2

x=6/2

x=3

3) state the optimal value

y=0.5(x+3)(x-9)

y=0.5(3+3)( 3-9)

y=0.5(6)(-6)

y=-18

Example 2: y=2(x+1)(x+4)

1) find the zeroes

y=2(x+1)(x+4)

0=2(x+1)(x+4)

X+1=0 and X+4=0

X=0-1 x=0-4

x=-1 x=-4

2) find the axis of symmetry

X=-1+-4/2

x=-5/2

x=-2.5

3) state the optimal value

y=2(x+-1)(x+4)

y=2(-2.5+1)(-2.5+4)

y=2(-1.5)(1.5)

y= -4.5

**GRAPHING FROM FACTORED FORM**

You need 3 key points to graph your parabola from your equation in factored form.

There are 4 main steps to graph your equation.

- Find the x-intercepts of the relation.
- Find the x-value of the vertex, which is the same as finding the axis of symmetry.
- Find the y-value of the vertex, which is the same as finding the optimal value.
- Now plot the 3 key points (two x-intercepts and the vertex) on the graph. Then connect the points and graph your parabola.

*Note that you might not always have two x-intercepts, there are cases where you have only 1 or NO x-intercept.

** 1. Finding the x-intercepts**

The example that we will be looking is :

**y = (x+4)(x+2)**

Take your equation and set y equal to 0.

You need to find to seperate solutions, therefore set each bracket equal to 0.

Solve ecah bracket.

You will get an answer that is x = # , write it as a point (X,0).

**2. Finding the x-value of the vertex**

The x-value of the vertex is the same as the axis of symmetry.

Use the formula (r+s)/2 to find the x-value.

You simply take the sum of the two x-intercepts that found before and divide it by 2.

**3. Finding the y-value of the vertex**

The y-value of the vertex is the same as the optimal value.

Take your original equation and replace the x with x-value of the vertex that we found.

Then solve for y

**4. Graphing the equation using the key points**

**Equation : y = (x+4)(x+2)**

Plot the 3 key points that we found through the above steps.

1. x-intercept #1 : (-4,0)

2. x-intercept #2 : ( 2,0)

3. Vertex : (-1,-9)

**REAL LIFE SITUATION INVOLVING FACTORED FORM **

The predicted flight path of a toy rocket used in a mathematics project is defined by the relation

h=-3(d-2) (d-12), where d is the horizontal distance, in metres, from a wall, and h is the height, in meters, above the ground.

a) Sketch a graph of the path of the rocket.

b) How far from the wall is the rocket when it is launched?

When it comes to questions that ask about the distance that an object started at, it is referring to the x-intercept. The distance the object started at will be the smallest number between the two x-intercepts. Therefore, according to this equation, the rocket was launched 2m away from the wall.

c) How far from the wall is the rocket when it lands on the ground?

When it comes to questions that ask about the distance that an object hits the ground at, it is referring to the x-intercept. The distance the object hits the ground at will be the larger number between the two x-intercepts. Therefore, according to this equation, the rocket lands on the ground 12m away from the wall.

d) What is the maximum height of the rocket, and how far, horizontally, is it from the wall at that moment?

This question is asking you to find the vertex. So, we will start off by solving for the axis of symmetry to get the distance and then substituting that value (the value of x) into the equation to get the height.

Therefore, the rocket will reach a maximum height of 75m when it is 7m away from the wall.

**CONVERTING FACTORED FORM INTO STANDARD FORM**

To convert the equation of a quadratic relation from factored form to standard form,

we will expand, regroup, then simplify the factored form.

**VIDEO: GRAPHING USING FACTORED FORM**

**Big Idea – find x int by setting y=0, and vertex is between the two x-ints**

## Multiplying Binomials

**WORD PROBLEM INVOLVING MULTIPLYING BINOMIALS**

1. A rectangle has width *w *centimetres and length 3 cm more than its width.

a) Draw a diagram of the rectangle.

The original formula for the area of a rectangle is a=(l)(w). Therefore, the formula to find the area is a= w(w+3).

c) Expand and simplify the area expression.

**Big Idea – can get from Factored form to Standard Form**

## Common Factors and Simple Grouping

Common factoring should *always* be the first thing you look for when you have an equation in standard form. You should look to see if each of the terms in the equation has a similar component that each of the terms can easily be divided into. The common factor can either be a variable (x) or a coefficient (any number.)

Example 1: 8x + 6

Step #1: Find GCF ( what is the greatest # in common between 8 and 6)

GCF=2

Step #2: Write solution with brackets

8x/2 + 6/2 = 4x + 3

= 2(4x + 3)

Example 2: 12x³ - 6x²

step #1: GCF= 6x²

Step #2: 6x²(2x-1)

check your answer!

6x²(2x-1)

=12x³ - 6x²

Example 4: 2x(x+4) -5(x+4)

Step #1: GCF= (x+4)

Step #2: (x+4) (2x-5)

link to check out: http://www.mathsisfun.com/algebra/factoring-quadratics.html

**WORD PROBLEM INVOLVING COMMON FACTORS AND SIMPLE GROUPING **

Write an expression, in factored form, for the shaded region.

a= 45x²- 15y²

The gcf of 45 and 15 is 15.

a= 15 (3x²- y²)

**Big Idea – Factoring will take us from Standard form back to Factored form**

## Simple trinomials

Simple trinomials have 3 terms, and can be written in the form y = ax² + bx + c

Expanding:

x² + 7x + 12= (x+3) (x+4)

Factoring: When factorign a polynomial of the form ax² + bx + c (when a=1), we want to find:

- Two numbers that ADD to give you the value of b
- Two numbers that MULTIPLY to give you the value of c

Example: Factor x² - 8x + 12

= (x-6) (x-2)

Check your Answer:

-6 + -2= -8

-6 x -2= 12

## NOTE

**REAL LIFE SITUATION INVOLVING SIMPLE TRINOMIALS **

*t*is time in seconds, and

*h*is the height, in metres.

a) Write the formula in factored form.

b)When will the rock hit the water?

In the form y= a (x-r) (x-s) the *r* and *s *values gives you the values of the x-intercepts but it needs to be changed to the opposite sign. The question asks for the distance at which the rock hits the water, so it is asking for the value of the second x-intercept. Therefore, the rock will hit the water in 5 seconds.

**Big Idea – Factoring will take us from Standard form back to Factored form**

## Complex Trinomials

*not*1. That makes it a little trickier, and more often the method

*trial and error*or

*guess and check*will have to be used.

## Note

**REAL LIFE SITUATION INVOLVING COMPLEX TRINOMIALS **

The height of a ball thrown from the top of a ladder can be approximated by the formula h=-2t²+4t+48, where t is the time, in seconds and h is the height, in metres.

a) Write the formula in factored form.

b) Determine when the ball will hit the ground.

In the form y= a (x-r) (x-s) the *r* and *s *values gives you the values of the x-intercepts but it needs to be changed to the opposite sign. The question asks for the distance at which the ball hits the ground, so it is asking for the value of the second x-intercept. Therefore, the ball will hit the ground in 6 seconds.

**Big Idea – Factoring will take us from Standard form back to Factored form**

## Perfect Squares

What is a perfect square?

Perfect Square Trinomial is the product of two binomials. But, both the binomials are same. When factoring some quadratics which gives identical factors, that quadratics are Perfect Square Trinomials.

In this section, you will learn to recognize trinomials of the form x² + 2ax + a² and factor them to (x+a)².

Example of a perfect square:

x² + 10x + 25

= (x + 5)(x + 5)

= (x + 5)²

Question 1: Factor the trinomial x² + 4x + 4

Solution:

Given x² + 4x + 4

The above trinomial can be written as x² + 2x + 2x + 4

Take x as common from first two terms,

x(x + 2) + 2x + 4

Take 2 as common from last two terms.

x(x + 2) + 2(x + 2)

(x + 2)(x + 2)

In this it is the product of two identical binomials

(x + 2)2 , which is a perfect square.

Question 2: Factor the trinomial x² + 8x + 16

Solution:

Given x² + 8x + 16

The above trinomial can be written as x² + 4x + 4x + 16

Take x as common from first two terms,

x(x + 4) + 4x + 16

Take 4 as common from last two terms.

x(x + 4) + 4(x + 4)

(x + 4)(x + 4)

In this, it is the product of two identical binomials

(x + 4)2, which is a perfect square.

**REAL LIFE SITUATION INVOLVING PERFECT SQUARES **

C= 2t² -24t +150, where* t *is the time, in months, the appliance is running. What is the minimum cost of running the appliance and when does this happen?

C= 2t² -24t +150

C= 2 (t² -12t) +150

C= 2 (t² -12t +36 -36) +150

C= 2 (t² -12t +36) -36 +150

C= 2 (t² -12t +36) -72+150

C= 2 (t² -12t +36) +78

C= 2 (t-6)² +78

In the form y= a (x-h)² +k, the h value gives you the x value of the vertex but you need to change it to the opposite sign and the k value gives you the y value of the vertex. Therefore, the minimum cost of running the appliance is $78 and this occurs when the appliance runs for 6 months.

**Big Idea – Factoring will take us from Standard form back to Factored form**

## Difference of Squares

When working with a difference of squares, you must check to see that:

- Each term is a squared term
- There is a
*minus*sign between the two terms - The two terms that are squared are separated by a subtraction sign like this: a² - b²

**REAL LIFE SITUATION INVOLVING A DIFFERENCE OF SQUARES **

A square has side length 8a. One dimension is increased by 3b and the other is decreased by 3b.

a) Find an algebraic expression for the resulting area. Expand and simplify.

Using the formula for the area of a square you will plug in the information provided to create a formula.

Given Information:

- Side length= 8a
- One side= +3b
- Another side= -3b

Therefore the equation is (8a+3b) (8a-3b).

a= (8a+3b) (8a-3b)

a= 64a²-9b²

b) Calculate the area of the square if a represents 4cm and b represents 2cm.

a= 64a²-9b²

a= 64 (4)²-9 (2)²

a=64 (16)-9 (4)

a= 1024- 36

a= 988

Therefore, the area of the square is 988cm².

**Big Idea – Factoring will take us from Standard form back to Factored form**

## RECAP

## COMPLETING THE SQUARE

## Note

## Examples

**REAL LIFE SITUATION INVOLVING COMPLETING THE SQUARES #1 **

1.The path of a ball is modelled by the equation y= -x² +2x +3, where *x* is the horizontal distance, in metres, from a fence and *y* is the height, in metres, above the ground.

a) What is the maximum height of the ball, and at what horizontal distance does it occur?

y= -x² +2x +3

y= -1(x² -2x) +3

y= -1(x² -2x +1- 1) +3

y= -1(x² -2x +1) -1+ 3

y= -1(x² -2x +1) +1+3

y= -1(x² -2x +1) +4

y= -1(x² -1)² +4

Therefore the maximum height of the ball is 4m and this occurs when the horizontal distance is at 1m.

**REAL LIFE SITUATION INVOLVING COMPLETING THE SQUARES #2**

Let r= the total revenue, in dollars

Let x= the number of $0.50 decreases in price

Revenue is the product of the price and the number rented.

r= (12 -0.5x) (36+ 2x)

r= (12 -0.5x) (36+ 2x)

r= 432+ 6x -x²

r= -x² +6x +432

r= -1(x² -6x) +432

r= -1(x² -6x +9 -9) +432

r= -1(x² -6x +9) -9 +432

r= -1(x² -6x +9) +9 +432

r= -1(x² -6x +9) +441

r= -1(x²-3)² +441

The relation reaches a maximum value of 441 when x=3.

There should be three price reductions of $0.50 to maximize the revenue.

12- 0.5 (3)= 10.50

Therefore, a price of $10.50 maximizes Alex's revenue.

**Big Idea – getting from standard form to vertex form**

## Solving Quadratic Relations

## Note

## Example

## Word Problem

## Standard form in algebra tiles

You can also use algebra tiles, to convert the same equation into standard form.

Think of this like a room with dimensions (x+2) by (x+1). The area of the room is (x+2)(x+1). We are trying to determine another way of expressing the area.

## Quadratic Formula

What is the quadratic formula?

· To solve a quadratic equation from standard form, we use the Quadratic formula.

· We could find the x-intercepts of the equation using this formula.

· We use this when it is hard to solve an equation by factoring.

Ex: When there are decimals

· We also use quadratic formula in situations where the equation doesn't factor at all.

The quadratic formula is used to solve a very specific type of equation, called a quadratic equation. These equations are usually written in the following form:

## Examples

**FINDING THE AXIS OF SYMMETRY**

Finding the axis of symmetry is the same as finding x-value of the vertex.

You can use a simple formula to find the axis symmetry when the equation is it's standard form.

You could also use the previous method that we learned but if you made a mistake in finding your x-intercepts then your vertex will also be wrong.

The formula is:

**EXAMPLE**

y = 2x² +3x -5

a = 2

b = 3

axis of symmetry : x = -b / 2a

= -(3) / 2(2)

= -3 / 4

= -0.75

**WORD PROBLEM **

Now let's look at a real life situation in which we could solve and equation given in standard form.

Knowing how to find the axis of symmetry, optimal value, vertex and x-intercepts will help you answer the questions from a word problem.

**VIDEO: GRAPHING INVOLVING FACTORED FORM**

## recap

## Connections

Throughout this unit I have notices that every topic that we learnt within the unit can be linked. For example, you can change vertex form to standard form and vice versa, factored form to standard form and vice versa, and vertex form to factored form. Now converting the equations into the 3 different forms can require either expanding, factoring, completing the square, using the quadratic formula, etc. In the process of these conversions you also will gather the necessary information to be able to graph the parabola of the equation (vertex, x-intercepts, axis of symmetry, etc.) A more specific example of this would be that through the equations we gather the information needed to graph the parabola, but in standard form you will not be able to gather any information by using the equation. So, you must convert it to either factor or vertex form to proceed with the procedure to graph the parabola. The quadratics unit is all linked together and knowing and understanding each topic and how they connect to each other will help you better understand the unit as a whole.