Representing Functions August 14, 2010

VocabularyBefore LessonAfter Lesson Previous Knowledge Dependent Variables Independent Variables Y-Intercept Roots or Functions Zeros New Concepts Maximum / Minimum Axis of Symmetry Quadratic Equation Vertex

 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?

 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?

 What do the Eiffel Tower, the Brooklyn Bridge, and the Sistine Chapel all have in common? Flicker.com Landscape-photo.net

a, b, c, r, s, h, k = constants a>0 : concave up, a minimum exists a<0 : concave down, a maximum exists

y =x 2 a=1  a>0 ∴Minimum exists at the vertex at point (0,0) y =-x 2 a=-1  a<0 ∴Maximum exists at the vertex at 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

 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 x=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?

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 = ( )/2 = 15 ∴she should increase the price by $45 for a total cost of $445.

Solution #2: P(x) = (100-x)(350+5x) = -5x x  standard form (a<0: maximum exists) = -5(x 2 -30x-7000) Use the complete the square method P(x) = -5(x 2 -30x ) = -5(x 2 -30x+225)-5( ) = -5(x-15)  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.

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.

VocabularyBefore LessonAfter Lesson Previous Knowledge Dependent Variables Independent Variables Y-Intercept Roots or Functions Zeros New Concepts Maximum / Minimum Axis of Symmetry Quadratic Equation Vertex

 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?