Which One Changes Faster?: IM Alg1.5.19
[url=https://im.kendallhunt.com/HS/teachers/1/5/19/preparation.html]“Which One Changes Faster?”[/url] from IM Algebra I © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg1.5.19 Lesson
- IM Alg1.5.19 Lesson: Which One Changes Faster?
- Alg1.5.19 Practice
- IM Alg1.5.19 Practice: Which One Changes Faster?
IM Alg1.5.19 Lesson: Which One Changes Faster?
Here is a graph.
Which equation do you think the graph represents? Use the graph to support your reasoning.[br][list][*][math]y=120+\left(3.7\right)\cdot x[/math][br][/*][/list][list][*][math]y=120\cdot\left(1.03\right)^x[/math][br][/*][/list]
What information might help you decide more easily whether the graph represents a linear or an exponential function?[br]
A family has $1,000 to invest and is considering two options:
[size=150]investing in government bonds that offer 2% simple interest, or investing in a savings account at a bank, which charges a $20 fee to open an account and pays 2% compound interest. Both options pay interest annually.[/size][br][size=150][br]Here are two tables showing what they would earn in the first couple of years if they do not invest additional amounts or withdraw any money.[/size][br]
Bonds: How does the investment grow with simple interest?
Savings account: How are the amounts $999.60 and $1,019.59 calculated?[br]
For each option, write an equation to represent the relationship between the amount of money and the number of years of investment.[br]
Which investment option should the family choose? Use your equations or calculations to support your answer.
Use graphing technology to graph the two investment options and show how the money grows in each.
Complete the table of values for the functions f and g.
[size=150][math]f(x)=2x[/math] and [math]g(x)=(1.01)^x[/math][/size]
Based on the table of values, which function do you think grows faster? Explain your reasoning.[br]
Which function do you think will reach a value of 2,000 first? Show your reasoning. If you get stuck, consider increasing [math]x[/math] by 100 a few times and record the function values in the table.[br]
Consider the functions [math]g\left(x\right)=x^5[/math] and [math]f\left(x\right)=5^x[/math]. While it is true that [math]f\left(7\right)>g\left(7\right)[/math], for example, it is hard to check this using mental math. Find a value of [math]x[/math] for which properties of exponents allow you to conclude that [math]f\left(x\right)>g\left(x\right)[/math] without a calculator.
IM Alg1.5.19 Practice: Which One Changes Faster?
Functions a, b, c, d, e and f are given below. Classify each function as linear, exponential, or neither.
Here are 4 equations defining 4 different functions, a, b, c and d.
[size=150]List them in order of increasing rate of change. That is, start with the one that grows the slowest and end with the one that grows the quickest.[br][br][table][tr][td][math]a(x)=5x+3[/math][/td][td][math]b(x)=3x+5[/math][/td][td][math]c(x)=x+4[/math][/td][td][math]d(x)=1+4x[/math][/td][/tr][/table][/size]
[size=150]Function [math]f[/math] is defined by [math]f\left(x\right)=3x+5[/math] and function [math]g[/math] is defined by [math]g\left(x\right)=\left(1.1\right)^x[/math][/size].[br]Complete the table below with values of [math]f\left(x\right)[/math] and [math]g\left(x\right)[/math]. When necessary, round to 2 decimal places. You may use the scientific calculator beneath the table to help with calculations.
Which function do you think grows faster? Explain your reasoning.
Use technology to create graphs representing f and g.
What graphing window do you have to use to see the value of [math]x[/math] where [math]g[/math] becomes greater than [math]f[/math] for that [math]x[/math]?
[size=150]Functions [math]m[/math] and [math]n[/math] are given by [math]m\left(x\right)=\left(1.05\right)^x[/math] and [math]n\left(x\right)=\frac{5}{8}x[/math]. As [math]x[/math] increases from 0:[/size][br]Which function reaches 30 first?[br]
Which function reaches 100 first?[br]
[size=150]The functions [math]f[/math] and [math]g[/math] are defined by [math]f\left(x\right)=8x+33[/math] and [math]g\left(x\right)=2\cdot\left(1.2\right)^x[/math].[br][/size][br][size=100]Which function eventually grows faster, [math]f[/math] or [math]g[/math]? Explain how you know.[/size]
Explain why the graphs of [math]f[/math] and [math]g[/math] meet for a positive value of [math]x[/math].[br]
[size=150]A line segment of length [math]\ell[/math] is scaled by a factor of 1.5 to produce a segment with length [math]m[/math]. The new segment is then scaled by a factor of 1.5 to give a segment of length [math]n[/math].[/size][br][br]What scale factor takes the segment of length [math]\ell[/math] to the segment of length [math]n[/math]? Explain your reasoning.
A couple needs to get a loan of $5,000 and has to choose between three options.
[list][*]Option A: [math]2\frac{1}{4}\%[/math] applied quarterly[/*][*]Option B: [math]3\%[/math] applied every 4 months[/*][*]Option C: [math]4\frac{1}{2}\%[/math] applied semi-annually[/*][/list]If they make no payments for 5 years, which option will give them the least amount owed after 5 years? Use a mathematical model for each option to explain your choice.