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QUADRATICS

INTRODUCTION TO QUADRATICS

Algebra 2

Real Life: Why would we ever need this?

■ Quadratics have a highest power of 2

■ Standard Form

• f(x) = ax2 + bx + c

• a, b, and c are constants

What Makes a Quadratic a Quadratic?

■ There are 4 ways to solve quadratic equations:

1. Taking Square Roots

2. Factoring

3. Quadratic Formula

4. Completing the Square

Ways to Solve Quadratics

■ When do we use this…..

– Whenever b is 0

– In other words, when you only have an x2

Solving Quadratics: Taking Square Roots

■ n2 = 16 ■ x2 = 80

Solving Quadratics: Taking Square Roots Examples

■ -3k2 = -132 ■ 9v2 = 531

Solving Quadratic Equations: Taking Square Root Examples

■ 8b2 + 7 = 15 ■ 5n2 + 6 = 91

Solving Quadratic Equations: Taking Square Root Examples

■ -10 – 9b2 = -874 ■ 4x2 + 9 = 41

Solving Quadratic Equations: Taking Square Root Examples

■ Taking Square Roots

Solving Quadratics: Taking Square Roots Homework

Bell Ringer

Solve the following quadratic equations.1. 8x2 -5 = 3

2. 6x2 + 7 = 277

FACTORING INTRODUCTION

Algebra 2

Ways to Solve Quadratics

■ What are the 4 ways to solve quadratics?

1. Taking Square Roots

2. Factoring

3. Quadratic Formula

4. Completing the Square

Solving Quadratics By Factoring

What is a factor?

• a number or quantity that when multiplied with another

produces a given number or expression

• Example: 2 is a factor of 12, 7 is a factor of 63

When do we factor quadratic equations?

■ When a is 1

Solving Quadratics by Factoring: Examples

■ x2 – 10x + 24■ x2 + 14x + 48

Solving Quadratics by Factoring: Examples

■ x2 + 11x + 30■ x2 + 2x – 15

Solving Quadratics by Factoring: Examples

■ x2 – 2x – 3 ■ x2 – 2x – 35

Solving Quadratics by Factoring: Homework

Solving Quadratics by Factoring Day

1

Bell Ringer

■ Factor each of the following quadratic expressions

1. x2 – 11x + 28

2. x2 + x - 30

SOLVING QUADRATIC EQUATIONS BY

FACTORINGAlgebra 2

Ways to Solve Quadratics

■ What are the 4 ways to solve quadratics?

1. Taking Square Roots

2. Factoring

3. Quadratic Formula

4. Completing the Square

Solving Quadratics By Factoring

What is a factor?

• a number or quantity that when multiplied with another produces a given

number or expression

• Example: 2 is a factor of 12, 7 is a factor of 63

When do we factor quadratic equations?

■ Easiest and most recognizable when a is

1

■ Will focus on when a isn’t 1

Solving Quadratic Equations by Factoring: Examples

■ Find the zeroes of the following quadratic equation.

x2 – 6x + 5 = 0

Solving Quadratic Equations by Factoring: Examples

■ Find the zeroes of the following quadratic equation.

x2 – x – 42 = 0

Solving Quadratic Equations by Factoring: Examples

■ Find the zeroes of the following quadratic equation.

3𝑝2 − 2𝑝 − 5 = 0

Solving Quadratic Equations by Factoring: Examples

■ Find the zeroes of the following quadratic equation.

3𝑛2 − 8𝑛 + 4 = 0

Solving Quadratic Equations by Factoring: Examples

■ Find the zeroes of the following quadratic equation.

5𝑛2 + 19𝑛 + 12 = 0

Solving Quadratic Equations by Factoring: Examples

■ Find the zeroes of the following quadratic equation.

9𝑘2 + 66𝑘 + 21 = 0

Solving Quadratic Equations by Factoring: Homework

Solving Quadratic Equations by

Factoring

Bell Ringer

Solve the following quadratic equations by factoring

1. 35x2 - 17x – 30 = 0

2. 6x2 + 49x + 8 = 0

SOLVING QUADRATIC EQUATIONS WITH THE QUADRATIC FORMULA

Algebra 2

Ways to Solve Quadratic Equations

■ What are the 4 ways to solve quadratic equations?

1. Taking Square Roots

2. Factoring

3. Quadratic Formula

4. Completing the Square

When do we use the Quadratic Formula?

■ Can always use the quadratic formula, but

it isn’t always the easiest method…

Solving Quadratic Equations using the Quadratic Formula

■ What is the standard form of a quadratic equation?

■ What is the quadratic formula?

Solving Quadratic Equations using the Quadratic Formula: Examples

■ 4x2 + 5x – 26 = 0 ■ 2r2 – 3r + 9 = 0

Solving Quadratic Equations using the Quadratic Formula: Examples

■ 4x2 + 6x + 8 = 0 ■ 5x2 + 8x + 16 = 0

Solving Quadratic Equations using the Quadratic Formula: Examples

■ 4x2 – 5x + 4 = 0 ■ 6x2 – 4x + 6 = 0

Solving Quadratic Equations using the Quadratic Formula: Homework

Solving Quad Equations: Quad Formula

Day 1

Bell Ringer

Solve the following quadratic equations using the

quadratic formula

1. 2x2 - 7x – 9 = 0

2. 3x2 + 2x + 11 = 0

SOLVING QUADRATIC EQUATIONS WITH THE

QUADRATIC FORMULA: DAY 2

Algebra 2

Solving Quadratic Equations with Quadratic Formula: Examples

■ 6x2 + 8x – 73 = -9 ■ x2 – 12x – 11 = -10

Solving Quadratic Equations with Quadratic Formula: Examples

■ 9x2 + 2x = –12 ■ 2x2 + 4 = –5

Solving Quadratic Equations with Quadratic Formula: Homework

Solving Quad Equations: Quad Formula

Day 2

Bell Ringer

Simplify the following radicals:1. 392

2. −98

3. −128

4. 112

SOLVING QUADRATICS BY

COMPLETING THE SQUARE

Algebra 2

4 Ways to Solve Quadratics

The 4 ways to solve quadratics are:

1. Taking Square Roots

2. Factoring

3. Quadratic Formula

4. Completing the Square

Solving Quadratics By completing the Square: When?

■ When do we complete the square to solve

quadratics:

– When b is even and a is 1

Solving Quadratics By Completing the Square: Steps

Steps to Completing the Square

1. Move all constants to the right (away from the letters)

2. Take b and cut it in half, then square that number

3. Add this number to both sides

4. Factor the left

5. Solve

Solving Quads By Completing the Square: Examples

■ x2 + 2x – 99 = 0

Solving Quads By Completing the Square: Examples

■ x2 + 4x – 30 = 0

Solving Quads By Completing the Square: Examples

■ x2 + 20x – 104 = –9

Solving Quads By Completing the Square: Examples

■ x2 – 18x – 43 = –6

Solving Quads By Completing the Square: Examples

■ x2 – 2x – 89 = 8

Solving Quads by Completing the Square: Homework

■ Solving Quads: Completing the Square

Worksheet

Bell Ringer – Completing the Square

𝑥2 + 30𝑥 − 9 = 0 𝑥2 + 14𝑥 + 84 = −14

Completing the Square

𝑛2 − 16𝑛 + 60 = 0 𝑏2 − 2𝑏 − 80 = 0

Completing the Square

𝑚2 + 2𝑚 − 8 = 0 𝑥2 + 20𝑥 − 37 = 0

Completing the Square

𝑥2 − 2𝑥 − 3 = 0 𝑥2 + 18𝑥 + 72 = 0

Completing the Square

𝑥2 − 12𝑥 − 28 = 0 𝑚2 − 8𝑚 + 7 = 0

Completing the Square

𝑝2 − 6𝑝 − 17 = 0 𝑚2 + 8𝑚 − 88 = 0

Completing the Square

𝑏2 − 20𝑏 − 71 = 6 𝑎2 − 2𝑎 − 36 = 6

Completing the Square

𝑝2 − 4𝑝 − 62 = 2 𝑝2 + 6𝑝 + 3 = 10

Completing the Square

𝑥2 − 8𝑥 − 42 = 9 𝑥2 − 10𝑥 − 94 = 9

QUADRATIC WORD PROBLEMS

Algebra 2

Solving Word Problems

■ Underline important info

■ Circle verbs/what being asked to do

■ Always ask yourself: What do they really want me to find?

Shortcut to Find Vertex

■ To find the x-coordinate of the vertex

• 𝑥 = −𝑏

2𝑎

– How would you find the y-coordinate of the vertex?

– Plug in the x-coordinate found

Word Problems

The height, h, in feet of an object above the ground is given by h = -16t2 + 64t + 190,

where t is the time in seconds.

a. Find the time it takes the object to reach its maximum height.

b. Find the maximum height of the object.

c. Find the time it takes the object to hit the ground.

Word Problems

■ The path of a rocket is given by the equation: ℎ = −16𝑡2 + 128𝑡, where h is the

height in feet of the rocket and t is the time in seconds after it is launched.

a. How long is the rocket in the air?

b. What is the maximum height the rocket reaches?

c. About how high is the rocket after 1 second?

Word Problems

A manufacturer of tennis balls has a daily cost of C(x) = 200 – 10x + 0.01x2, where

C is the total cost in dollars and x is the number of tennis balls produced. What

number of tennis balls will produce the minimum?

Word Problems

The value of Jon’s stock portfolio is given by the function v(t) = 50 + 77t+ 3t2,

where v is the value of the portfolio in hundreds of dollars and t is the time in

months. How much money did Jon start with? What is the minimum value of Jon’s

portfolio?

Word Problems

Find the number of units that produce the maximum revenue, R = 900x − 0.1x2,

where R is the total revenue (in dollars) and x is the number of units sold.

QUADRATIC GRAPHS AND THEIR PROPERTIES

Vocabulary■ A quadratic function is a type of nonlinear function that models certain

situations where the rate of change is not constant.

■ The graph of a quadratic function is a symmetric curve with the highest

or lowest point corresponding to a maximum or minimum value.

■ Standard Form of a Quadratic Function:

– A quadratic function is a function that can be written in the form

𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, where a ≠ 0. This form is called the standard

form of a quadratic function.

– Examples: 𝑦 = 3𝑥2 𝑦 = 𝑥2 + 9 𝑦 = 𝑥2 − 𝑥 − 2

Vocabulary

■ The simplest quadratic function 𝑓 𝑥 = 𝑥2 𝑜𝑟 𝑦 = 𝑥2 is the quadratic parent function.

■ The graph of a quadratic function is a U-shaped curve called a parabola.

■ You can fold a parabola so that the two sides match exactly. This property is called symmetry.

■ The fold or line that divides the parabola into two matching halves is called the axis of symmetry.

Vocabulary

■ The highest or lowest point of a parabola is its vertex, what is on

the axis of symmetry.

■ If a > 0 in y = ax2 + bx + c, the parabola opens upward. The vertex

is the minimum point, or lowest point, of the parabola.

■ If a < 0 in y = ax2 + bx + c , the parabola opens downward. The

vertex is the maximum point, or highest point, of the parabola.

Identifying a Vertex

PracticeIdentify the vertex.

Practice

Vocabulary

■ You can use the fact that a parabola is symmetric to graph it quickly.

■ First, find the coordinates of the vertex and several points on one side of the

vertex.

■ Then reflect the points across the axis of symmetry.

■ For graphs of functions of the form y = 𝑎𝑥2, the vertex is at the origin.

■ The axis of symmetry is the y–axis, or x = 0.

Graphing 𝑦 = 𝑎𝑥2

Graph the function. Make a table of values. What are the domain and range?

1. 𝑦 =1

3𝑥2

2. 𝑦 = −3𝑥2

3. 𝑦 = 4𝑥2

Practice

Graph each function. Then identify the domain and range of the function.

1. 𝑦 = −4𝑥2

2. 𝑦 = −1

3𝑥2

3. 𝑓 𝑥 = 1.5𝑥2

4. 𝑓 𝑥 =2

3𝑥2

Vocabulary

■ The y – axis is the axis of symmetry for graphs of functions

of the form 𝑦 = 𝑎𝑥2 + 𝑐.

■ The value of c translates the graph up or down.

Graphing 𝑦 = 𝑎𝑥2 + 𝑐

What is the relationship of the following graphs?

1. 𝑦 = 2𝑥2 + 3 𝑎𝑛𝑑 𝑦 = 2𝑥2

2. 𝑦 = 𝑥2 𝑎𝑛𝑑 𝑦 = 𝑥2 − 3

3. 𝑦 = −1

2𝑥2 𝑎𝑛𝑑 𝑦 = −

1

2𝑥2 + 1

Practice

Graph each function.

1. 𝑓 𝑥 = 𝑥2 + 4

2. 𝑓 𝑥 = −𝑥2 − 3

3. 𝑓 𝑥 =1

2𝑥2 + 2

Vocabulary

As an object falls, its speed continues to increase, so its height above the ground decreases at a faster and faster rate.

Ignoring air resistance, you can model the object’s height with the function h = –16t2 + c.

The height h is in feet, the time t is in seconds, and the object’s initial height c is in feet.

Using the Falling Object Model

An acorn drops from a tree branch 20 feet above the ground. The function h = –16t2 + 20 gives the

height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At what

time does the the acorn hit the ground?

t h = –16t2 + 20

Using the Falling Object Model

An acorn drops from a tree branch 70 feet above the ground. The function h = –16t2 + 70 gives the

height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At what

time does the the acorn hit the ground?

t h = –16t2 + 70

PracticeA person walking across a bridge accidentally drops an orange into the

river below from a height of 40 feet. The function ℎ = −16𝑡2 + 40 gives

the orange’s approximate height h above the water, in feet, after t

seconds. In how many seconds will the orange hit the water?

A bird drops a stick to the ground from a height of 80 feet. The function

ℎ = −16𝑡2 + 80 gives the stick’s approximate height h above the

ground, in feet, after t seconds. Graph the function. At about what time

does the stick hit the ground?

QUADRATIC FUNCTIONS

Vocabulary■ In the quadratic function y = ax2 + bx + c, the value of b affects the position of the

axis of symmetry.

■ The axis of symmetry changes with each equation because of the change in the b-

value. The equation of the axis of symmetry is related to the ratio 𝑏

𝑎.

■ The equation of the axis of symmetry is 𝑥 = −1

2

𝑏

𝑎𝑜𝑟 𝑥 =

−𝑏

2𝑎.

■ Graph of a Quadratic Function

o The graph of y = ax2 + bx + c, where a ≠ 0, has the line 𝑥 =−𝑏

2𝑎as its axis of symmetry. The x–

coordinate of the vertex is −𝑏

2𝑎.

Vocabulary

■ When you substitute x = 0 into the equation y = ax2 + bx +

c, you get y = c. So the y–intercept of a quadratic function is

c.

■ You can use the axis of symmetry and the y–intercept to

help you graph a quadratic function.

Graphing y = ax2 + bx + c

What is the graph of the function? Show the axis of symmetry.

1. 𝑦 = 𝑥2 − 6𝑥 + 4

2. 𝑦 = −𝑥2 + 4𝑥 − 2

3. 𝑦 = 2𝑥2 + 3

4. 𝑦 = −3𝑥2 + 12𝑥 + 1

5. 𝑓 𝑥 = 𝑥2 + 4𝑥 − 5

6. 𝑓 𝑥 = −4𝑥2 + 11

Practice

What is the graph of the function? Show the line of symmetry.

1. 𝑦 = 2𝑥2 − 6𝑥 + 1

2. 𝑓 𝑥 = 2𝑥2 + 4𝑥 − 1

3. 𝑦 = 6𝑥2 + 6𝑥 − 5

4. 𝑓 𝑥 = −5𝑥2 + 3𝑥 + 2

5. 𝑦 = −2𝑥2 − 10𝑥

6. 𝑦 = −4𝑥2 − 16𝑥 − 3

Vocabulary

■ You have used h = –16t2 + c to find the height h above the ground

of an object falling from an initial height c at time t.

■ If an object projected into the air given an initial upward velocity v

continues with no additional force of its own, the formula h = –16t2

+ vt + c givens its approximate height above the ground.

Using a Vertical Motion Model

During halftime of a basketball game, a sling shot launches T–shirts at the crowd. A T–shirt launched with an initial upward velocity of 72 feet per second. The T–shirt is caught 35 feet above the court. The T–shirt is launched from a height of 5 feet.

a. How long will it take the T–shirt to reach its maximum height?

b. What is the maximum height?

c. What is the range of the function that models the height of the T–shirt over time?

Using a Vertical Motion Model

During halftime of a basketball game, a sling shot launches T–shirts at the crowd. A T–shirt launched with an initial upward velocity of 64 feet per second. The T–shirt is caught 35 feet above the court. The T–shirt is launched from a height of 5 feet.

a. How long will it take the T–shirt to reach its maximum height?

b. What is the maximum height?

c. What is the range of the function that models the height of the T–shirt over time?

Practice

A baseball is thrown into the air with an upward velocity of 30 feet per second. Its

height h, in feet, after t seconds is given by the function h = –16t2 + 30t + 6.

a. How long will it take the ball to reach its maximum height?

b. What is the ball’s maximum height?

c. What is the range of the function?