Quadratic Functions

A Quadratic Function is an equation that has the form

2 cxaxy b

The graph of a Quadratic Equation is a u-shaped curve called a Parabola.The Vertex is the highest or lowest point of the parabola.

If a is positive, the parabola opens upward, and the vertex is the minimum point.

If a is negative, the parabola opens downward, and the vertex is the maximum point.

Maybe we should look at

some diagrams.

The Vertex is also called the Turning Point.

Maximum and Minimum Points

2 42y x x

a is positive, therefore the parabola opens upward, and the vertex is the minimum point.

22 4xy x

a is negative, therefore the parabola opens downward, and the vertex is the maximum point.

VertexTurning Point

(1, -2)

VertexTurning Point

(-1, 2)

Axis of SymmetryThe Axis of Symmetry of a parabola is the line that splits the parabola in half lengthwise. The Axis of Symmetry always goes through the Vertex of the parabola.

Let’s look at some graphs.

2 42y x x

Axis ofSymmetry x = 1

22 4xy x

Axis ofSymmetry

x = -1

Finding the Axis of Symmetry

You can find the Axis of Symmetry of any quadratic equation by using the formula 2

xab

Let’s take a look at those equations again.

22( ) 4x xf x

a = 2, b = -4, c = 0

2( )

2 24

( )ba

1x

To find the coordinates of the vertex, plug the value of x into the original function .

21 1( ) 1( ) ( )2 4f 2

vertex (1, -2)

22( ) 4x xf x

a = -2, b = -4, c = 0

2( )

2 24

( )ba

1x 22( ) (1 14( )1)f 2

vertex (-1, 2)

Using a Graphing Calculator to find the Axis of

Symmetry2 782x xy

First, let’s use the formula to find the axis of symmetry algebraically.

2( )

2 2 )2

(8

ab

x

Now let’s use the calculator to find the axis of symmetry graphically.

Enter the equation in Y1 of the Y = window.

View the graph by pressingGRAPH

2( ) ( ) 82 (2 2 7)2f

vertex (2, -1)

Press 2nd CALC 3(minimum)

Move the curser slightly to the left of the vertex and press ENTER

Move the curser slightly to the right of the vertex and pressENTER

Press ENTERThe calculator calculates the coordinates of the vertex.

Now push the

easy button.

Graphing a Quadratic Equation Using a Table of

Values2 64x xy Grap

hin the interval

1 5x

x y

-1 -1

0 -6

1 -9

2 -10

3 -9

4 -6

5 -1

Axis ofSymmetry

x = 2

Vertex (2, -10)

2x

ab

4( )(12 )

x

42

x

2x

That was easy

Graphing

Examples

x y

Quadratic Functions Homework

Page 156:1 – 4, 6

Answer all questions on the graph paper.

Show all your work

2) 43 x

4) 0 6x

Solving Quadratic Equations

When you solve a quadratic equation, the x values that you calculate are referred to as the roots of the equation.

2 cxaxy b When a quadratic function is in the form of , the roots can be found by setting the equation equal to zero and solving.When a quadratic equation is factorable, then it can be solved algebraically.

This is actually pretty easy. Let’s look at some

examples.

Sometimes, the roots of the equation are referred to as the solution set.

Factoring and SolvingQuadratic Equations

Find the solution set of the following quadratic functions.

228 3x x Rewrite the equation in ax2 + bx + c = 0 format.

2 28 03x x

Rewrite the equation so that a is positive.

2 083 2x x

Factor the equation.4) 07( )(x x Set each factor equal to zero and solve.

7 0x 4 0x 7x

Write the solution set.7 4{ , }x

4x

2 40 13x x

2 40 013x x

2 4 013 0x x

5) 08( )(x x 8 0x 5 0x

8x 5x

8{ }5, x

More Factoring ExamplesFor what values of x is the following fraction undefined?

2

2

112

xx x

If the denominator was equal to zero, the fraction would be undefined.

2 012x x

4) 03( )(x x

or3 4x x

The fraction would be undefined at

3 4{ , }x

Solve the following equation for x

32 5x x

x

Cross-multiply.2 5 2 6x x x

Rewrite the equation in ax2 + bx + c = 0 format.

2 6 07x x Factor the equation.

1) 06( )(x x

or6 1 x x

6{ }1, x

Solving Quadratic Equations by Graphing

Find the roots of the following quadratic function.

2 10 3x x

Let’s solve by factoring first.

2 003 1x x 2) 05( )(x x

5 2{ , }x Now let’s solve by graphing.

1) Enter the equation in Y1 .

2) View the graph by pressingGRAPH

3) Press 2nd CALC 2 (zero)

4) Move the curser slightly to the left of the vertex and press ENTER

5) Move the curser slightly to the right of the vertex and press

ENTER

6) Press ENTER

The calculator calculates the 1st root.

Repeat steps 3 – 6 to calculate the 2nd root.

More Solving QuadraticEquations by Graphing

Approximate the roots of the following quadratic function to the nearest hundredth.2 2 3x x Set equation equal to zero.

2 2 03x x

This equation is unfactorable, so we have to use our calculator.

1) Enter the equation in Y1 .

2) View the graph by pressingGRAPH

3) Press 2nd CALC 2 (zero)

4) Move the curser slightly to the left of the vertex and pressENTER

5) Move the curser slightly to the right of the vertex and press

ENTER

6) Press ENTER

The calculator calculates the 1st root.

Repeat steps 3 – 6 to calculate the 2nd root.

.5615528x

3.5615528x

Round off your answer. 0.56{ , 56}3.x

Solving Quadratic Equationswith no Middle Term

22 0 05x

Since there is no middle term, this equation is unfactorable.

22 50x 2 25x

2 25x

or5 5 x x

}5{x

Let’s check with our calculator to make sure the roots are

correct.Approximate the roots of the following quadratic function to the nearest hundredth.23 0 06x

23 60x 2 20x

2 20x

or4.47 4.47 x x 4 7{ }.4x

Let’s check with our calculator to make sure the roots are

correct.

That was easy

Linear Quadratic Systems

1x y 2 34x xy

Algebraically

1x y 1y x

2 34x xy 2 4 3 1x x x

2 45 0x x 2 4 05x x

1) 04( )(x x 4x 1x

1 4y 3y

(4, )3

1 1y 0y

(1,0)

Graphically

Y1 = 1 xY2 =

2 4 3x x View the graph by pressingGRAPH

Press 2nd CALC 5 (intersect)

Press

ENTER ENTER ENTER

The calculator calculates the first point of intersection.(1,0)Repeat the same process . Be sure to move the curser closer to the second point before pressing enter.(4, )3

Projectile MotionA ball is thrown in the air so that its height, h, in feet after t seconds is given by the equation 21144 6t th a. Find the number of seconds that the ball is in the air when it reaches a height of 128 feet.

b. After how many seconds will the ball hit the ground?

21144 6t th 2144128 16t t

2 144 1 816 2 0t t 2 8 09t t

1) 08( )(t t

8t 1t The ball reaches 128 feet at 1 second and at 8 seconds.

21144 6t th 2164 01 4t t

2 1416 04t t 2 146 01 4t t

1 416 0( )4t t 0t 16 144t

9t

The ball hits the ground after 9 seconds.

Maybe we should check this on our calculator.

More Projectile MotionA model rocket is launched from ground level. At t seconds after it is launched, it is h meters above the ground, where 24.9( 6) 68.t th t

What is the maximum height, to the nearest meter, attained by the model rocket?

I know how to do this.

We need to find the maximum height. So, first we’ll find the axis of symmetry, then use that x-value to find the corresponding y-value.

2x

ab

2( .8.6

)64 9

7 27 7( ) ( ) ( )764.9 68.h

240.1( )7 480.2h 7 2( ) 40.1h The maximum height is

approximately 240 meters.

Maximizing the Area of a Rectangle

Stanley has 30 yards of fencing that he wishes to use to enclose a rectangular garden. If all the fencing is used, what is the maximum area of the garden that can be enclosed?

Let x represent the length and let w represent the width.

Since all the fencing must be used, the perimeter of the garden will be 30 yards, and we can use the following equation.2 2 30x w

15x w 15w x

x

15 - x

Let A(x) represent the area of the rectangle.

( )() )( 15xA x x 2( ) 15 xA x x

2( ) 15x xA x

The maximum value occurs at

2x

ab

2 )151(

7.5

27.5 7.5( ( ) ( )7 5) 15 .A

( ) 56.257.5 112.5A

( )7.5 56.25A

The maximum area is 56.25

yards2.

Quadratic Equations Homework

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