A Constant Rate of Change Name Part 1

Transcription

1 A Constant Rate of Change Name Part 1 Consider the function table below. Complete this page by solving the problems at the bottom. Use a separate sheet of paper for your descriptions and explanations. x y n Recursive Formula Explicit Formula Sketch graph below: Slope Constant Rate of Change Constant Added (to get from one y-value to the next) 1. Examine the function table above and briefly describe any patterns you see. 2. Look down the column of y-values in the table. If NOW is the y-value at a particular place in the table and NEXT is the y-value in the next row down, write an equation using NEXT and NOW that describes the pattern in the y-values. Write this equation in the box next to the down arrow. 3. Using the additional column beside the column of y-values, rewrite the y-values using the number 4 and/or the number in your equation from #2. Describe the pattern you found.

2 4. Using the last row of the table on the previous page, generalize the pattern with an expression used to find the y-value of the function if the x-value is n. (Hint: Use #3 and that column as help.) 5. Using the expression above, write an equation in the form y=.. showing the relationship between x and y. Write this equation on the previous page in the box above the right-arrow. 6. What type of function is represented by the table and by the equations you have found? Describe the basic characteristics of this function. 7. Sketch a small graph of this function in the location shown on the previous page. Slope is a fundamental feature of the graph of a linear function. What is the slope of this function? 8. The slope of a graph of a linear function also shows the constant rate of change of y with respect to x. Describe how the table shows this constant rate of change. 9. Examine the equations in the boxes on the previous page. a. Describe how the slope and constant rate of change are shown in the two equations. b. Circle the number in those equations that corresponds to the slope. To show what the circled number represents, draw an arrow from each circled number to the box at the bottom of the page, and enter the number in the box. c. Do you think that one equation shows the slope and constant rate of change more clearly than the other? Explain.

3 A Constant Multiplier Name Part 2 Consider the function table below. Your task is to complete the page by solving the problems below. Use a separate sheet of paper for your descriptions and explanations. When you are finished you will compare this work with the work that you did in Part 1. x y n Explicit Formula Recursive Formula Sketch a graph below: Constant Multiplier 1. Examine the function table above and briefly describe any patterns you see. 2. Look down the column of y-values in the table. If NOW is the y-value at a particular place in the table and NEXT is the y-value in the next row down, write an equation using NEXT and NOW that describes the pattern in the y-values. Write this equation in the box next to the down arrow. 3. Using the additional column beside the column of y-values, rewrite the y-values using the number 4 and/or the number in your equation from #2. Describe the pattern you found. 4. Using the last row of the table on the previous page, generalize the pattern with an expression used to find the y-value of the function if the x-value is n. (Hint: Use #3 and that column as help.)

4 5. Using the expression above, write an equation in the form y=.. showing the relationship between x and y. Write this equation on the previous page in the box above the right-arrow. 6. What type of function is represented by the table and by the equations you have found? Describe the basic characteristics of this function. 7. Sketch a small graph of this function in the location shown on the previous page. Does the graph have a constant slope? Is there a constant rate of change with respect to x? Explain. 8. A fundamental characteristic of exponential functions is that there is a constant multiplier (but not a constant rate of change of y with respect to x.) Describe how the table shows this constant multiplier. 9. Examine the equations in the boxes on the previous page. a. Describe how the constant multiplier is shown in the two equations. b. Circle the number in those equations that corresponds to the constant multiplier. To show what the circled number represents, draw an arrow from each circled number to the box at the bottom of the page, and enter the number in the box. c. Do you think that one equation shows the constant multiplier more clearly than the other? Explain.

5 10. Compare the NEXT-NOW equations that you wrote in part 1 and part 2. These equations are recursive formulas, since they describe one value in terms of the previous value. a. Describe how these two NEXT-NOW equations are similar and how they are different. b. How do the y= equations that you wrote in part 1 and part 2 show this difference? c. How do the graphs of the two functions show this difference? d. How do the tables for the two functions show this difference? 11. Write a paragraph describing linear and exponential functions. Compare function types and describe a contextual situation that could represent each.