F2S Name: ___________________________ Problem Set #2 Due: Friday, October 17, 2014
Investigating Graphs of Exponential Functions Using Desmos
𝒚 = 𝒂𝒃𝒙 𝒂𝒏𝒅 𝒚 = 𝒂𝒃𝒙!𝒉 + 𝒌
Objectives:
• Explore the graph of the exponential function • Explore how the values of a, b, h, and k change the graph of the exponential function.
To get started:
• Go to Desmos.com • Click on “Launch Calculator” in the large red box • Click on the drop down menu at the top left corner, left of “Untitled Graph” • Choose the first item “New Blank Graph”
1. Investigate the base of the exponential function
Type: 𝑓 𝑥 = 𝑏! and add the slider for b
a) How does the value of b affect the graph of the function?
b) Does the point (0, 1) belong to the graph for all values of b? _____________
c) Describe how the graph of 𝑓 𝑥 = 2! and 𝑓 𝑥 = !!
! differ.
d) If b > 1, the graph rises/falls to the right (circle one)
If 0 < b < 1, the graph rises/falls to the right (circle one)
e) Are the output values of f(x) ever negative?
Asymptotes: The graph hugs a line called the asymptote. The asymptote is always at y = 0 (x-‐axis) unless you move the parent graph up or down.
2. Investigate the effect of a in 𝒚 = 𝒂 ∙ 𝒃𝒙. Type 𝑔 𝑥 = 𝑎 ∙ 𝑏! and add the slider for a
a) let a = 1 and b = 2 , so that 𝑔 𝑥 = 2! (let this be the “parent function”)
Identify two key points on the graph: ____________________________________
b) What effect does “a” have on the graph of 𝑔 𝑥 = 2! ? (use the slider to change a)
c) let a = 1 and 𝑏 = !! , so that 𝑔 𝑥 = !
!
! (let this be the “parent function”)
Identify two key points on the graph: ____________________________________
d) What effect does “a” have on the graph of 𝑔 𝑥 = !!
! ? (use the slider to change a)
3. Investigate 𝒇 𝒙 = 𝒃𝒙!𝒉. Conjecture: 𝒇 𝒙 = 𝟐𝒙!𝟑 is the same as 𝒇 𝒙 = 𝟏𝟖𝟐𝒙
a) Graph 𝑓 𝑥 = 2! and 𝑓 𝑥 = 2!!!.
Describe what the exponent (𝑥 − 3) does to the graph of 𝑓 𝑥 = 2!
b) Graph 𝑓 𝑥 = 2!!! and 𝑓 𝑥 = 𝑎𝑏! and add sliders for a and b. Let 𝑎 = !! and 𝑏 = 2
Can you verify graphically that these are the same?
Can you explain algebraically why these are the same graphs?
4. Investigate the effect the value of k has on the graph of 𝒇(𝒙) = 𝒃𝒙 + 𝒌
Graph 𝑓 𝑥 = 𝑏! + 𝑘 and add the slider for k. Let 𝑏 = 2 and vary the value of k with the slider. Describe how k changes the function.
5. Investigating 𝒚 = 𝒆𝒙
One mathematical constant of significance is the number 𝑒 ≈ 2.718281828. The function 𝑔 𝑥 = 𝑒! is just one exponential function among many, but it shows up in so many contexts that we call it the natural exponential function.
Graph 𝑦 = 𝑒! and 𝑦 = 𝑎! Notice that the graph of each exponential function 𝑦 = 𝑎! is related to the graph of 𝑦 = 𝑒! by some stretch factor.
Which function grows fastest as x-‐values get large?
𝑎) 𝑦 = 2! b) 𝑦 = 𝑒! c) 𝑦 = 5! d) 𝑦 = 3!
Problem Set #2 Name:_______________________________________
Homework -‐ Show all work on separate paper. Finish for HW Due Saturday 10/18
1. Do this activity in class (Friday, 10/17) using Excel (see detail below) Suppose you make a deal with your parents that rather than take your regular monthly allowance of $50, you will accept just $.01 on the first day of the month and so on, with the condition that they double the amount they give you each day. Your parents rapidly figure out that in one week they will have given you just $1.27, ($0.01 + $0.02 + $0.04 + $0.08 + $0.16 + $0.32 + $0.64) so they agree. Will you get more than $50 that month? If so, on what day of the month will you have exceeded $50? Can you figure out how much money they will owe you on the last day of the month? (Assume there are 30 days in the month.)
a) Note: this is an exponential function with base 2 – calculate equation “by hand”
b) Calculate an exponential equation for this data using exponential regression
c) Excel detail –
• make a column for day, amount, sum of days • make a chart of daily allowance (y) vs. day (x) and get an exponential trendline
equation • make a chart of total allowance (y) vs. day(x)
2. Populations
a) A tree frog population doubles every three weeks. Suppose that currently, there are 10 tree frogs in your back yard. How many tree frogs will there be in six months, assuming that there are four weeks each month?
b) How long will it take this population to be 10,240?
3. Tennis Tournament – Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds?