Lesson. The Vertex Activity 1 The Vertex 1. a. How do you know that the graph of C œ ÐB Ñ ' is a parabola? b. Does the parabola open up or down? Why? c. What is the smallest C-value on the graph of C œ ÐB Ñ '? (Hint: What is the smallest value that ÐB Ñ can have?) d. Which B-value gives us the smallest C-value? Why? e. Graph the equation in the "friendly" window and verify your answers.. a. Find the vertex of the graph of C œ $ÐB Ñ '. b. Write the equation in part (a) in standard form.. a. Find an equation for a parabola whose vertex is Ð%ß (ÑÞ b. Sketch your parabola at right. c. Find an equation for the parabola in part (a) if one point on the graph is (,*) Þ 10 9 8 1 109 8 1 0 1 8 910 10 98 d. Sketch the parabola from part (c) on the same grid.
Activity Vertex Form The batter in a softball game hits the ball when it is feet above the ground. The ball reaches the greatest height, feet, directly above the head of the left-fielder, who is 00 feet from home plate. a. The path of the ball is shown at right. Label the C-intercept and the vertex with their coordinates. b. Write an equation for the height of the ball in terms of the horizontal distance it has traveled. c. Find the height of the ball when it reaches the left field wall, which is feet from home plate. If the wall is 10 feet tall, did the batter hit a home run (did the ball go over the wall)? Activity Maximum or Minimum Value A farmer plans to fence a rectangular grazing area along a river using 00 yards of fence. a. Draw a sketch illustrating the grazing area, and label its length and width. b. Let A stand for the width of the rectangle, and write an expression for the length in terms of the width. c. Write an equation for the area E of the grazing land in terms of A. d. Graph the equation. Thousands of s quare yards 1 11 10 9 8 1 0 0 0 0 80 100 10 10 10
e. Use algebra to answer the questions: What is the largest area the farmer can enclose? What are the dimensions of the largest area? Activity Revenue The local theater group sold tickets to its opening night performance for $ and drew an audience of 100 people. The next night they reduced the ticket price by $0. and 10 more people attended; that is, 110 people bought tickets at $. apiece. In fact, for each $0. reduction in ticket price, 10 additional tickets can be sold. a. Complete the table. Number of Price Reductions Price of Ticket Number of Tickets Sold Total Revenue! &Þ!! "!! &!! " %Þ(& ""! &Þ&! $ & ) "! "" b. Let B represent the Number of price reductions, as in the first column of the table. Write algebraic expressions in terms of B for: The Price of a Ticket after B price reductions: Price œ The Number of Tickets Sold at that price: Number œ The total revenue from ticket sales: Revenue œ
c. Enter your expressions into the calculator: Y" œ the price of a ticketß Y œ the number of tickets soldß Y$ œ the total revenue. Use the Table feature to verify that your algebraic expressions agree with your table from part (a). d. Use your calculator to graph your equation for total revenue in terms of B. Sketch the graph on the grid. e. What is the maximum revenue possible from ticket sales? What is the value of B at that revenue? 00 0 00 0 00 0 00 0 00 1 10 1 100 0 01 8 9 101111111118190 f. Use your formulas for Y" and Y to answer the questions: What price should the theater group charge for a ticket to generate the maximum revenue? How many tickets will they sell at that price? 8
Wrap-Up In this Lesson, we worked on the following skills and goals related to quadratic models: Find the vertex of a parabola Find the maximum or minimum value of a quadratic model Write a quadratic equation in vertex form by completing the square Find an equation for a parabola given the vertex and one other point Write a quadratic model for revenue Check Your Understanding 1. The height of a golf ball in meters is given by œ %Þ*>!>Þ a. Explain how to find out when the ball hits the ground. b. Explain how to find the greatest height the ball reaches.. Explain why you need another point besides the vertex to find the equation of a parabola.. Explain why revenue does not increase indefinitely as you increase price.. What is the smallest value that C can have if C œ ÐB $Ñ '? What value of B produces that smallest C-value? 9