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Possible Equation for a Function on a Graph

5.12.4 Possible Equation for a Function on a Graph

5.12.4 • Possible Equation for a Function on a Graph

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Examine the three graphs representing three exponential functions: f f , g g , and h h .

Line graph showing three upward-curving lines labeled a (blue), b (green), and c (orange) on axes labeled x (horizontal) and y (vertical), with values increasing rapidly as x increases. h and g intersect on the y-axis and f is below both of them with a wider curve. h is a steeper curve than g. The lines are h, g, and then f across the graph.

The functions f f and h h are defined as:

  • f ( x ) = 10 · 2 x f ( x ) = 10 · 2 x
  • h ( x ) = 20 · 4 x h ( x ) = 20 · 4 x
1.

Which of the following could define function g g ? Be prepared to show your reasoning.

  1. g ( x ) = 20 · ( 1.5 ) x g ( x ) = 20 · ( 1.5 ) x

  2. g ( x ) = 20 · ( 2.5 ) x g ( x ) = 20 · ( 2.5 ) x

  3. g ( x ) = 10 · ( 3.5 ) x g ( x ) = 10 · ( 3.5 ) x

  4. g ( x ) = 20 · ( 4.5 ) x g ( x ) = 20 · ( 4.5 ) x

g ( x ) = 20 · ( 2.5 ) x g ( x ) = 20 · ( 2.5 ) x .

2.

Explain your reasoning.

Compare your answer:

The graph of g g  has the same y y -intercept as the graph of h h , which is 20. It grows more quickly than f f  but more slowly than h h , so the growth factor must be greater than 2 but less than 4.