University:
Augusta University
Course:
MATH 4800 | Secondary Mathematics from an Advanced Perspective
Academic year:

2024
Views:

193

Pages:

2
Author:

Autumn Cox

Question 1 Tell whether the function represents exponential growth or exponential decay. Identify the percent increase or decrease. Then graph the function. f(x) = (1/3)^x Answer Step 1 We are given the function: (f(x) = (1/3)^x) Step 2 We identify the value of the base. The base, (1/3) is greater than 0 and smaller than 1, the function represents exponential decay. Step 3 We make a table of values: x | y ---|--- -2 | 9 -1 | 3 0 | 1 1 | 1/3 2 | 1/9 Step 4 We plot the points from the table. Then draw a smoth curve from right to left, that begins just above the x-axis, passes throigh the plotted points and moves up to the left. Step 5 The percent of decrease is: (1 - 1/3 = 2/3 = 66%) Question 2 Determine whether the function represents exponential growth or exponential decay. Then identify the percent rate of change. y = 10(1.07)^x Answer If the base of the exponential function is smaller than 1, then the function represents exponential decay. If the base is greater than 1, then the function represents exponential growth. For this function we note that the base is 1.07(> 1) and thus the function represents exponential growth. The percentage rate of change is the (positive) difference between 100% and the base: |100% - 1.07| = |100% - 107%| = |-7%| = 7% Question 3 In the exponential growth or decay function y = y0e^kt, what does y0 represent? What does k represent? Answer In the growth and decay function that is y = y0e^kt, y0 represents the initial quatity, and k represents the rate of growth or decay. Question 4 Find the exponential function f(x) = Ca^x whose graph has the given points (0,2) and (3,54). Answer f(0) = 2 = Ca^0 C = 2 f(3) = 54 = 2a^3 27 = a^3 a = sqrt(27) a = 3 f(x) = 2(3^x) Question 5 A table for a function f is given to the right. (a) Determine whether function f represents exponential growth, exponential decay, or linear growth. (b) Find a formula for f. x | -2 | -1 | 0 | 1 | 2 ---|--- f(x) | 1.28 | 3.2 | 8 | 20 | 50 Answer a) f(x) represents exponential growth (decay) according as (f(x+1))/f(x) = r>1(<1). It represents linear growth if f(x + 1) - f(x) = constant ∀x Now f(-1)/f(-2) = 3.2/1.28 = 5/2 Also f(0)/f(-1) = f(1)/f(0) = f(2)/f(1) = 5/2 b) Since the ratio is a constant with r > 1, f represents exponential growth. Let f(x) = A*(5/2)^x f(0) = A = 8 Hence f(x) = 8 * (5/2)^x