Maximum or Minimum of a Quadratic Function

Representing FunctionsAugust 14, 2010

Vocabulary Before Lesson After Lesson

Previous Knowledge

Dependent Variables

Independent Variables

Y-Intercept

Roots or Functions Zeros

New Concepts

Maximum / Minimum

Axis of Symmetry

Quadratic Equation

Vertex

Today’s Vocabulary

How do assigning mathematical relations to the physical world improve our understanding of the world?

How does assigning these relations improve our understanding of the unknown?

How do understanding functions and relations inspire further research and development?

Essential Questions

Get into groups of 4 Task

◦ You have 200 feet of fencing to enclose an area up against a building wall

◦ The area is to be enclosed on 3 sides◦ Layout the fenced area using the Lego blocks provided◦ Calculate the area of the yard for each step◦ Fill in the provided table for Length, Width, and Area

Reflection◦ Note the shape and the Area as you go through the

exercise.◦ What dimensions create the largest enclosed area?

Manipulative Exercise

What do the Eiffel Tower, the Brooklyn Bridge, and the Sistine Chapel all have in common?

Connection Question?

Flicker.comLandscape-photo.net

Maximum and Minimum of Quadratic Equations

a, b, c, r, s, h, k = constants

a>0 : concave up, a minimum existsa<0 : concave down, a maximum exists

Maximum and Minimum of Quadratic Equations

y =x2

a=1 a>0∴Minimum exists at the vertexat point (0,0)

y =-x2

a=-1 a<0∴Maximum exists at the vertexat point (0,0)

Standard Form – multiply the entire equation and collect like terms will result in the standard form

Factored Form (Root Form) - The standard form can be factored if a perfect factor exists. If no perfect factor exists, then partial factoring can be used to located the vertex.

Vertex Form – The standard form can be manipulated by using the complete the square method

Techniques to Change Forms for a Represented Quadratic Equation

The axis of symmetry goes directly through the middle of the parabola.

It is located at the average of the two roots

Axis of Symmetry

2

srh

Axis of Symmetryx=h

Problem:Kamilla sells wedding cakes and would like to know how to

maximize her profits. She sells on average 100 cakes per year. She sells them for

$400 per cake. The cost to make the cake, not including labor, is $50. To increase her profits, she must sell the cakes at a higher price, but she will end up losing customers. For every $5 increase, she loses 1 customer. How much should she raise the prices to maximize her profits?

Using Quadratic Equations to Solve Real Life Problems

Solution #1:Define the independent variable. Let x be the number of incremental price adjustments.

Profit = (total cakes sold) x (profit from one cake) P(x) = (100-x)(350+5x)

=-5(x-100)(x+70) factored form (a<0: maximum exists)

Roots exist at x=100, x=-70

Vertex position is at x = (-70+100)/2 = 15

∴she should increase the price by $45 for a total cost of $445.

Using Quadratic Equations to Solve Real Life Problems

Solution #2: P(x) = (100-x)(350+5x)

= -5x2+150x+35000 standard form (a<0: maximum exists)

= -5(x2-30x-7000)Use the complete the square method

P(x) = -5(x2-30x+225-225-7000)= -5(x2-30x+225)-5(-225-7000)= -5(x-15)2+36125 vertex form (a<0: maximum exists)

∴ Vertex exists at the point (15,36125)

∴ Kamilla should raiser her price by 15x5 = $45 to receive maximum profit of $36125.

Using Quadratic Equations to Solve Real Life Problems

Using Quadratic Equations to Solve Real Life Problems

-120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

Kamilla’s Profits

Using Quadratic Equations to Solve Real Life Problems

-120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

f(x) = − 5 x² + 150 x + 35000

Kamilla’s Profits

Problem:Medaille College would like to open a day care center. An

alumni has donated 200 feet of fencing to enclose the area up against the main wing. Medaille would like to know the maximum area that can be enclosed with the fencing.

a. Calculate the maximum area?b. What if the width could be no greater than 40 feet? What

is the maximum area?c. What is the width could be no greater than 60 feet? What

is the maximum area?

Use Excel to plot and verify your answers.

Lego Example

Vocabulary Before Lesson After Lesson

Previous Knowledge

Dependent Variables

Independent Variables

Y-Intercept

Roots or Functions Zeros

New Concepts

Maximum / Minimum

Axis of Symmetry

Quadratic Equation

Vertex

Journal – Today’s Vocabulary

How do assigning mathematical relations to the physical world improve our understanding of the world?

How does assigning these relations improve our understanding of the unknown?

How do understanding functions and relations inspire further research and development?

Journal - Essential Questions