Introduction to Quadratic Functions Connecting Your Knowledge

Transcription

1 Unit 4: Frogs, Fleas, & Painted Cubes//Investigation 1//Connections Name Class Date Introduction to Quadratic Functions Connecting Your Knowledge I can write equations for quadratic functions. Math / 30 Reflection / 10 Total / 40 Points For this packet, your goal is to earn will be from the math problems. 10 will be from the reflection section. Look at the point values of the following questions and answer the questions that you feel most comfortable to answer. Use the Distributive Property to write the expression in expanded form. Then, simplify. 1. 1(5 + 6) Expanded: Simplified:. (35 + 1) Expanded: Simplified: 3. 1(10 ) Expanded: Simplified: 3. 9(3 5) Expanded: Simplified:

2 Use the Distributive Property to write the expression in factored form Factored Form: Factored Form: Factored Form: Factored Form: Solve each equation x = x = 15 9x x = 85 5x

3 5 1. a. Extend the graph to include lengths greater than 3 meters to show areas for rectangles with a fixed perimeter. b. Make a table of data from the graph. Length Area c. What is the maximum area for a rectangle with this perimeter? What are the dimensions of this rectangle? Eduardo s neighborhood association subdivided a large rectangular field into two playing fields as shown in the diagram. a. Write TWO expressions to show two different ways you can calculate the area of the large field. EXPRESSION 1: EXPRESSION : b. Use the diagram and your two expressions in part (a) to explain the Distributive Property.

4 5 14. The equation p = d(100 d) gives the monthly profit p a photographer will earn if she charges d dollars for each print. a. Make a table for this equation. Dollars $0 $10 $0 $30 $40 $50 $60 $70 $80 $90 $100 Profit b. Make a graph for this equation. c. What price will produce the maximum profit? A rectangular soccer field has a perimeter of 400 yards. The equation L = 00 W represents the relationship between the length (L) and width (W) of the field. a. Explain why the equation for the length (L = 00 W) is correct. b. Is the relationship between length and width a quadratic function? Explain why or why not. c. Suppose the width of the soccer field is 50 yards. What is the length of the soccer field and what would the area be?

5 A beach has a rectangular swimming area for toddlers. One side of the swimming area is the shore. Buoys and a rope with a length of 0 meters are used to form the other three sides. a. How should you arrange the rope to make a rectangle with the maximum area? What would that maximum area be? b. In Problem 1.1 (in your Investigation packet) a fixed length of 0 meters was also used to form a rectangle. Compare the rectangle with a maximum area in that problem with the rectangle with a maximum area in this problem. Are the shapes and areas the same? Explain why or why not. Shapes: Areas: c. Make a graph relating the length and area for the possible rectangular swimming areas. d. How does this graph compare with the graph you drew in Problem 1.1?

6 Mr. DeAngelo is designing a school building. The music room floor will be a rectangle with an area of 1,00 square feet. a. Make a table showing 10 different possible combinations of lengths and widths that would create that area. Length Width b. Add a column to your table and fill out the perimeter of each rectangle. c. What patterns do you see in the perimeter column? What kinds of rectangles have large perimeters and what kinds have small perimeters? d. Write an equation you can use to calculate the length of the floor for any given width Consider rectangles with a perimeter of 90 meters and a side length (L). a. Express the other dimension of the rectangle in terms of L. b. Write an equation for the area in terms of L.

7 c. Make a graph of your equation. d. Use your equation to find the area of the rectangle with a length of 1 meters. SHOW YOUR WORK! e. How could you find the area in part (d) by using your graph? f. How could you find the area in part (d) by using your table? g. What is the maximum area possible for a rectangle with a perimeter of 90 meters? What are the dimensions of this rectangle?

8 Intro. to Quadratic Functions: Reflection In this investigation, you looked at the relationship between length and area for rectangles with a fixed perimeter. You learned that this relationship is a quadratic function. Answer the following questions in complete sentences. (This reflection is worth 10 of your total of 40.) 1. Describe the characteristics of graphs and tables of quadratic functions you have observed so far.. How do the patterns in a graph of a quadratic function appear in a table? 3. Describe two ways to find the maximum area for rectangles with a fixed perimeter. 4. Complete the following organizer outlining differences between quadratic functions and linear and exponential functions (you may use visual representation here). Table Graph Equation Linear Exponential Quadratic