Distinguish Linear and Exponential Growth

5.14.1 Distinguish Linear and Exponential Growth

5.14.1 • Distinguish Linear and Exponential Growth

Warm Up

Here is a graph.

Line graph with x-axis labeled 0 to 7 and y-axis labeled 0 to 200. The graph shows a steadily increasing line starting just below 125 on the y-axis and ending at about 150 at x = 7.
1.

Which equation do you think the graph represents? Use the graph to support your reasoning.

  • y = 120 + ( 3.7 ) · x y = 120 + ( 3.7 ) · x
  • y = 120 · ( 1.03 ) x y = 120 · ( 1.03 ) x

Compare your answer:

Your answer may vary, but here are some samples.

  • The graph is for y = 120 + 3.7 x y = 120 + 3.7 x because it appears linear. On the graph, when x = 7 x = 7 , the value of y y appears to be a little less than 150. When x = 7 x = 7 , 120 + ( 3.7 ) · 7 = 145.9 120 + ( 3.7 ) · 7 = 145.9 , so the slope of the line could be 3.7.
  • The graph is for y = 120 · ( 1.03 ) x y = 120 · ( 1.03 ) x . Multiplying 100 by 1.03 seven times, or 100 · ( 1.03 ) 7 100 · ( 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 y y -coordinate values for different values of x x using both equations, and they more or less agree with what is on the graph.
2.

What information might help you decide more easily whether the graph represents a linear or an exponential function?

Compare your answer:

Your answers may vary, but here are some examples:

  • Exact coordinates of points on the graph with whole-number x x -values
  • A larger graphing window showing larger domain values so that the behavior of the function can be more visible