Introduction In order to fully understand how various functions model real-world contexts, we need to understand how changing parameters will affect the functions. This lesson will explore the effect of changing parameters on linear and exponential functions. We will also interpret the effects of changing these parameters in a context : Interpreting Parameters

Key Concepts An exponential function is a function that may be written in the form f(x) = b x + k. A linear function is a function in the form f(x) = mx + b. The parameter of a function is a term that determines a specific form of a function but not the nature of the function. For a linear function written in slope-intercept form, f(x) = mx + b, the parameters are m and b. m represents the slope and b represents the y-intercept. Changing either of these parameters will change the function and the graph of the function : Interpreting Parameters

Key Concepts, continued For an exponential function in the form f(x) = b x + k, where b is a positive integer not equal to 1 and k can equal 0, the parameters are b and k. b is the growth factor and k is the vertical shift, or the number of units the graph of the function is moved up or down. Changing either of these parameters will change the function and the graph of the function : Interpreting Parameters

Key Concepts, continued Graphing Equations Using a TI-83/84: Step 1: Press [Y=]. Step 2: Key in the equation using [X, T, , n] for x. Step 3: Press [WINDOW] to change the viewing window, if necessary. Step 4: Enter in appropriate values for Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl, using the arrow keys to navigate. Step 5: Press [GRAPH] : Interpreting Parameters

Key Concepts, continued Graphing Equations Using a TI-Nspire: Step 1: Press the home key. Step 2: Arrow over to the graphing icon (the picture of the parabola or the U-shaped curve) and press [enter]. Step 3: Enter in the equation and press [enter]. Step 4: To change the viewing window: press [menu], arrow down to number 4: Window/Zoom, and click the center button of the navigation pad : Interpreting Parameters

Key Concepts, continued Step 5: Choose 1: Window settings by pressing the center button. Step 6: Enter in the appropriate XMin, XMax, YMin, and YMax fields. Step 7: Leave the XScale and YScale set to auto. Step 8: Use [tab] to navigate among the fields. Step 9: Press [tab] to “OK” when done and press [enter] : Interpreting Parameters

Common Errors/Misconceptions not understanding the difference between variables and parameters in a function mistaking the slope for the y-intercept or the y-intercept for the slope forgetting to identify the vertical shift in the context of an exponential problem when reading a word problem, not being able to identify the parameters in the context of the problem : Interpreting Parameters

Guided Practice Example 1 You visit a pick-your-own apple orchard. There is an entrance fee of $5.00, plus you pay $0.50 for each apple you pick. Write a function to represent this scenario. Complete a table of values to show your total cost if you pick 10, 20, 30, 40, and 50 apples. Graph the line and identify the parameters in this problem. What do the parameters represent in the context of the problem? : Interpreting Parameters

Guided Practice: Example 1, continued 1.Write a function. This scenario is represented by a linear function. Identify the slope and the y-intercept. The slope is the $0.50 charged for each apple picked. The y-intercept is the entrance fee of $5.00. Substitute the slope and the y-intercept into the linear function f(x) = mx + b, where m is the slope and b is the y-intercept. The function for this scenario is f(x) = 0.5x : Interpreting Parameters

Guided Practice: Example 1, continued 2.Create a table. Let x = the number of apples picked and f(x) = the total cost. Use the values 0, 10, 20, 30, 40, and 50 for x : Interpreting Parameters x0.5x + 5f(x)f(x) 00.5(0) (10) (20) (30) (40) (50) + 530

Guided Practice: Example 1, continued 3.Graph the function. Use the table of values to plot the equation of the line : Interpreting Parameters

Guided Practice: Example 1, continued 4.Identify the parameters. The parameters in this problem are the slope and the y-intercept. In this problem, the y-intercept is the entrance fee, $5.00, and the slope is the cost per apple, $ : Interpreting Parameters ✔

: Interpreting Parameters Guided Practice: Example 1, continued 13

Guided Practice Example 2 You deposit $100 into a long-term certificate of deposit (CD) in which your money will double every 7 years. Write a function to show how much money you will have in total in 7, 14, 21, 28, and 35 years. Use the function to create a table, and then graph the function. What do the parameters represent in the context of this problem? : Interpreting Parameters

Guided Practice: Example 2, continued 1.Write a function. This scenario is represented by an exponential function. The initial deposit is $100. Your money doubles every 7 years, so the growth factor is 2. The time period is 7 years. Substitute these values into the exponential function. The function for this scenario is : Interpreting Parameters

Guided Practice: Example 2, continued 2.Create a table. Let x = the number of years and f(x) = the amount of money in dollars. Use the values 7, 14, 21, 28, and 35 for x : Interpreting Parameters

Guided Practice: Example 2, continued : Interpreting Parameters xf(x)f(x)

Guided Practice: Example 2, continued 3.Graph the function. Use the table of values to plot the function : Interpreting Parameters

Guided Practice: Example 2, continued 4.Identify the parameters. The parameters in this problem are the starting amount of $100 and the base of : Interpreting Parameters ✔

: Interpreting Parameters Guided Practice: Example 2, continued