Lesson 3. Intercepts and Factors Activity 1 A Typical Quadratic Graph a. Verify that C œ ÐB (ÑÐB "Ñ is a quadratic equation. ( Hint: Expand the right side.) b. Graph C œ ÐB (ÑÐB "Ñ in the friendly window Xmin œ *Þ% Xmax œ *Þ% Ymin œ "! Ymax œ "! Sketch the graph on the grid. c. Use the Trace feature to locate the B-intercepts of the graph. d. Solve the equation ÐB (ÑÐB "Ñ œ! 3 1 0 1 3 5 10 1 e. Explain how your answers to parts ( c) and part ( d) are related. Activity X-Intercepts 1. a. Graph all three equations in the standard window, and sketch the graphs in the space at the right. What do you notice about the B-intercepts? (1) C œ B B "& () C œ $ÐB B "&Ñ (3) C œ!þðb B "&Ñ Multiplying a quadratic expression by a constant does not change the B -intercepts of the graph. b. Solve by factoring:!b %!B (!! œ! 5
. a. Graph C œ!"ðb. *B $'!Ñ on your calculator, using the ZInteger setting. b. Locate the B-intercepts of the graph. c. Use the B-intercepts to write the quadratic expression in factored form. (Do not try to factor the expression!) Activity 3 Perimeter and Area Do all rectangles with the same perimeter, say 3 inches, have the same area? 1. a. Sketch a rectangle with perimeter 3 inches and base 10 inches, and label its dimensions. (To find the height of the rectangle, reason as follows: The base plus the height makes up half of the rectangle's perimeter.) What is its area? b. Sketch a rectangle with perimeter 3 inches and base 1 inches, and label its dimensions. What is its area?. The table shows the bases of various rectangles with perimeter 3 inches. Fill in the height and the area of each. 3. Plot the points with coordinates ( Base, Area). (We will not use the heights of the rectangles.) Connect your data points with a smooth curve.. What is the largest area you found among rectangles with perimeter $' inches? What is the base for that rectangle? What is its height? Area: Base Height Area "! ) )! " ' ( $ "% & "( * "" % "' "& " ' ) "$ ( Base: Height:
5. Let B represent the base of the rectangle. Write an expression for the height of the rectangle in terms of B. ( Hint: If the perimeter of the rectangle is $' inches, what is the sum of the base and the height?) Height: Write an expression for the area of the rectangle in terms of B. Area: Activity Modeling A rancher has 30 yards of fence to enclose a rectangular pasture. If she uses a riverbank to border one side of the pasture, she can enclose 1,000 square yards of land. What will the dimensions of the pasture be then? We'll solve this problem in three different ways. First, make a sketch of the pasture: 1. Using a Table of Values Notice that you only have 30 yards of fence. So if you decide how wide the pasture is going to be, the length of the pasture is determined. See the table for two examples. a. Use the perimeter of the pasture, 30 yards, to write a formula for the length Width Length Area of the pasture in terms of its width. "! $%! $%!! (Be careful: remember that one side of the pasture does not need any! $! '%!! fence!) Length œ b. Make a table that shows the areas of pastures of various widths. c. Continue the table until you find the pasture whose area is 1,000 square yards. 7
. Using a Graph On your sketch of the pasture in part (1), label the width of the pasture as B. a. Write an expression for the length of the pasture if its width is BÞ ( Hint: how did you compute the length of the pasture in part (1)? Length œ b. Write an expression for the area E of the pasture if its width is B. Area œ c. Graph the equation for E on your calculator, using the window Xmin œ! Xmax œ ")) Ymin œ! Ymax œ "(!!! Then copy the graph onto the grid. d. Use the TRACE feature to find the pasture of area 1,000 square yards. Label the correct point on your graph. Thousands of square yards 0 1 1 1 1 10 0 0 0 0 0 100 10 10 10 10 00 3. Using an Equation a. Refer to steps (a) and (b) of part, and write an equation for the area E of the pasture in terms of its width B. Area œ b. Substitute E œ "'ß!!! and solve your equation algebraically.
Which of the three methods do you prefer? Why? Wrap-Up In this Lesson, we worked on the following skills and goals related to quadratic models: Solve quadratic equations by factoring Find a quadratic equation with given solutions Find the B-intercepts of a parabola Solve problems involving perimeter and area or the Pythagorean theorem Check Your Understanding 1. Delbert says that the solutions of the equation %ÐB ÑÐB "Ñ œ! are B œ %ß B œ ß and B œ ". Do you agree? Why or why not?. If the perimeter of a rectangle is 5 inches, and its width is A inches, write an expression for its length. 3. How can you use a graph to factor a quadratic expression?. In part of Activity, what did the two variables on the graph represent? What happened to the second variable as you increased the first variable? 9