MTH 103 Group Activity Problems (W1B) Name: Types of Functions and Their Rates of Change Section 1.4 part 1 (due April 6) Learning Objectives Identify linear and nonlinear functions Interpret slope as a rate of change Use and interpret average rate of change Linear Function Definition (found on page 42 in your textbook): Constant Function Definition (found on page 43 in your textbook): Recognizing linear and constant functions in symbolic form: Determine whether each function is a linear function or a nonlinear function. Briefly explain each one. For each function that is a linear function, additionally determine whether or not it is a constant function. 1. f(x) = 3x 2. g(x) = 2x 5 3. h(x) = 5 4. j(x) = 2x+8 4 5. k(x) = π Recognizing linear and constant functions in graph form: 6. Explain in words how you can recognize that a graph represents a linear function, and then draw two examples.
7. Explain in words how you can recognize that a graph represents a constant function, and then draw two examples. 8. Explain in words how you can recognize that a graph represents a nonlinear function, and then draw two examples. Recognizing linear and constant functions in verbal form: 9. Consider the situation: A driver starts 280 miles from home and drives toward home at a steady speed of 50 miles per hour. a. What characteristics of this situation allow it to be modeled with a linear function? b. The driver s distance from home is a function of time since they started driving. Define t, including units. t: Define D(t), including units. D(t): c. Write a formula for the linear function that models this situation. d. Create a table of values for this function e. Draw an accurate graph of the function. t D(t)
10. Let y represent the number of years since 1995 and let q(y) represent the number of quarters in one dollar during that year. a. What characteristics of this situation allow it to be modeled with a constant function? b. Write y or q(y) in each blank. Our independent variable is. Our dependent variable is. c. Write a formula for the constant function that models this situation. d. Create a table of values for this function e. Draw an accurate graph of the function. y q(y) 11. Consider the situation: A driver starts 280 miles from home and drives toward home. The driver starts fast but begins to have car trouble and drives slower and slower the closer they get to home. What characteristics of this situation DO NOT allow it to be modeled with a linear function? Calculating and interpreting slope of a line: Slope of a Line -- Figure 1.68 from page 43 in your textbook:
12. Calculate the slope of each linear relation whose graph is shown below. a. b. c. d. 13. Consider the graphs of the four linear relations for which you calculated slope. One graph rose from left to right, one graph went down from left to right, one graph was a constant linear function, and one graph was a vertical line (not a function). Separately for each of these four types of linear graphs, write a summary below of what will always be true about the slope value. Linear graph that rises from left to right (increasing linear function): Linear graph that goes down from left to right (decreasing linear function): Constant linear graph (horizontal line graph, constant linear function): Vertical line graph (linear relationship that is not a function):
14. The median age of the U.S. population for each year t between 1970 and 2010 can be approximated by the formula A(t) = 0.243t 450.8. a. Compute A(1980) and A(2000). Explain in words what your answers represent in the situation. b. State the value of the slope of the graph of A. Explain in words, including units, what the slope represents in the situation. 15. The number of times a cricket chirps in one minute is dependent upon the temperature. The number of chirps per minute at temperature F degrees Fahrenheit can be found using the formula C(F) = 1.6F 4, for 32 F 110. a. How many times will a cricket chirp in one minute when the temperature is 90 degrees Fahrenheit? Write your answer using function notation. b. State the value of the slope of the graph of C. Explain in words, including units, what the slope represents in the situation. Average Rate of Change: If a function is not linear, then it will not have the same slope (rate of change) everywhere. We can calculate the average rate of change between two points for nonlinear functions. Copy the equations from the Definition and the Note (found on page 51 in your textbook):
16. Find the average rate of change of each function below from x = -4 to x = -2 and also from x = 1 to x = 6. For each average rate of change that you calculate, label the starting and ending point of the interval on the graph and draw the line between the two points. a. f(x) = 0.3x 2 4 b. g(x) = x + 5 + 2 c. h(x) = 0.1x 3 2x + 1 17. A cylindrical tank contains 100 gallons of water. A plug is pulled from the bottom of the tank and the amount of water in gallons remaining in the tank after x minutes is given by A(x) = 100 (1 x 5 )2. a. Calculate the average rate of change of A from x = 1 to x = 1.5 and also from x = 2 to x = 2.5. Explain both of your results in the context of the situation, including units in your explanation. b. Compare the two average rates of change that you calculated. Are the same or different? Explain why.