2.2B Change in linear and exponential functions
What is the general form of an exponential function? Explain each component of this form.
What is the general form of a geometric sequence? Explain each component of this form.
Explain the relationship between exponential functions and geometric sequences.
How can an exponential function be expressed based on a known ratio and a point?
Explain how the domains of sequences and their corresponding functions may differ.
For the following exponential functions:
Express each function in geometric sequence general form.
Identify the the initial value.
Identify the constant proportion.
For the following geometric sequences:
Express the sequences as exponential functions.
Identify the initial value.
Identify the growth or decay rate.
Consider the geometric sequence g_n = 5 (2)^n represented by exponential function f \left( x \right)=5 (2)^x.
State the domain of the geometric sequence.
State the domain of the exponential function.
Discuss the similarities and differences in their domains.
Express the following exponential functions in the form f(x) = y_i r^{(x-x_i)} based on a known ratio, r, and a point, (x_i, y_i):
f \left( x \right) = 12(3)^x, point (1,36)
f \left( x \right)=25(0.4)^x, point (2,10)
f \left( x \right)= 15(1.5)^x, point (3,33.75)
f \left( x \right)=20(0.6)^x, point (4,4.32)
Given the following pair of points:
Determine if the function that goes through the points is exponential or not.
If exponential, determine exponential function.
(2,6) and (3,12)
(1,3) and (3,9)
(-2,4) and (2,1)
(-7,30) and (1,3)
Consider the following sequences.
State if the sequence is geometric.
If so, identify the constant ratio.
Construct the corresponding exponential function in the form g_n=g_0r^{(n-1)}.
3, 6, 12, 24, 48, \ldots
5, 10, 20, 45, \ldots
2, 6, 18, 72, 360, \ldots
6, 3, 1.5, 0.75, \ldots
Consider the geometric sequence 6, 24, 96, 384 \ldots and the exponential function f(x) = 6(4)^x.
Write the 4th term of the geometric sequence and the value of the exponential function when x=4. State if the results are same or not.
Write down the next three terms from the given of the geometric sequence and calculate the corresponding values of the exponential function. Compare your results.
A geometric sequence is defined by the exponential function f(x) = 3(5)^x.
What is the 5th term of the geometric sequence and the value of the exponential function when x=4?
If the common ratio of the geometric sequence and the base of the exponential function are doubled, how does this affect the 5th term of the geometric sequence and the value of the exponential function when x=4?
An exponential function is given by f(x) = 4(3)^x.
Represent the function in the general form of geometric sequence.
Find the 6th term of the geometric sequence and the value of the exponential function when x=5.
If the initial value of the geometric sequence and the coefficient of the exponential function are tripled, how does this affect the 6th term of the geometric sequence and the value of the exponential function when x=5?
The amount of a certain drug in a patient's bloodstream decreases over time, which can be modeled by an exponential function. Suppose the amount of drug decreases by a factor of 0.5 every 10 hours.
Write an exponential function to model the decay of this drug over time.
Determine the amount of drug left in the patient's bloodstream after 30 hours if the initial dose was 20 \text{ mg}.
How does the rate of decay affect the amount of drug left in the bloodstream over time?
A new viral video is being shared on social media and its views are increasing exponentially. The video got 100 views in the first hour and 400 views by the end of the 3rd hour. Assume the growth can be modeled by an exponential function of the form f(x) = a b^x.
Determine the model of the growth of the viewers over time.
Use this model to predict the number of video views at the end of the 5th hour.
Discuss the limitations of this model and the factors might affect the accuracy of your prediction?
An investment account with an initial balance of \$5000 is earning interest at an annual rate of 5\% compounded continuously. This scenario can be modeled by the exponential function f(t) = P (2.5)^{(rt)}, where P is the initial amount, r is the interest rate, and t is time in years.
Use the model to predict the account balance after 10 years.
What assumptions are made in this model?
Discuss the implications if these assumptions do not hold true.