Which One Changes Faster?

5 min

Narrative

At the beginning of this unit, students compared linear and exponential growth. They return to this comparison in this lesson. This Warm-up aims to show that, visually, it could be very difficult to distinguish linear and exponential growth for some domain of the function. While any exponential function eventually grows very quickly, it also can look remarkably linear over a portion of the domain.

Student Task

Here is a graph.

<p>Graph of line, origin O. Horizontal axis 0 to 7, by 1’s, labeled x. Vertical from 0 to 150, by 50’s, labeled y. Starts at 0 comma 120, appears to pass through 1 point 5 comma 125 and 7 comma 150.<br>  </p>

  1. Which equation do you think the graph represents? Use the graph to support your reasoning.
    •  y=120+(3.7)⋅xy=120 + (3.7)\boldcdot x
    •  y=120⋅(1.03)xy = 120 \boldcdot (1.03)^x
  2. What information might help you decide more easily whether the graph represents a linear or an exponential function?

Sample Response

  1. All answers are acceptable if backed up by a reasonable explanation. Sample responses:
    • The graph is for y=120+3.7xy=120+ 3.7x, because it appears to be linear. On the graph, when x=7x = 7 the value of yy appears to be a little less than 150. When x=7x = 7, 120+(3.7)⋅7=145.9120 + (3.7) \boldcdot 7 = 145.9 so the slope of the line could be 3.7.
    • The graph is for y=120⋅(1.03)xy=120 \boldcdot (1.03)^x. Multiplying 120 by 1.03 seven times, or 120⋅(1.03)7120 \boldcdot (1.03)^{7}, gives approximately 147.6, which more or less concurs with what is on the graph.
    • It is hard to tell. I tried finding the yy-coordinate values for different values of xx using both equations and they more or less agree with what is on the graph.
  2. Sample responses:
    • Exact coordinates of points on the graph with whole-number xx-values
    • A larger graphing window showing larger domain values so that the behavior of the function can be more visible

Synthesis

Invite students to share the rationales for their decision and their ideas for improving the clarity of the graph.

Help students understand that the graph of a linear function always looks like a line regardless of the domain in which it is plotted. An exponential function, however, can sometimes look linear, depending on the domain and range. Graphs are very helpful for seeing the general behavior of a function but not always for determining what kind of function is being graphed.

Consider showing students this image of the graph showing a larger domain and range as well as the original window (the red rectangle in the lower left).

<p>A graph.</p>

Or show a dynamic graph of y=120⋅(1.03)xy=120\boldcdot(1.03)^x starting with a small window in which the graph looks linear and then zooming out until the curve is visible.

Standards
Addressing
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
  • F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
  • F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
  • F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
  • F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
  • F-IF.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <em>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</em>
  • HSF-IF.B.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. <span>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.</span>
  • HSF-IF.B.5·Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <span>For example, if the function <span class="math">\(h(n)\)</span> gives the number of person-hours it takes to assemble <span class="math">\(n\)</span> engines in a factory, then the positive integers would be an appropriate domain for the function.</span>

15 min

15 min