Interpreting Features of an Exponential Graph

5.13.4 Interpreting Features of an Exponential Graph

5.13.4 • Interpreting Features of an Exponential Graph

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Here are two graphs representing the function $f$ given by $f (x) = 10 \cdot 2^{x}$ and the function $g$ defined by $g (x) = a \cdot b^{x}$ .

A graph with two exponential curves, f and g. Line g has three labeled points: (0, 3), (1, 12), and (2, 48). The points are marked with black dots. The x- and y-axes are shown. Line f has a large y-intercept than line g and intersects with line g.

In function $g$ , is $b$ greater than or less than 2? Explain how you know.

Compare your answer:

$b$ is greater than 2.

The graph of $g$ starts off being below the graph of $f$ , but later it surpasses the graph of $f$ . Since $g$ grows more quickly than $f$ , the growth factor $b$ for $g$ must be greater than the growth factor for $f$ , which is 2.
The growth factor from the first point $(0, 3)$ to the second point $(1, 12)$ is 4 because $\frac{12}{3} = 4$ , and 4 is greater than 2.

Write an equation that defines $g$ . Explain your answer.

Compare your answer:

$g (x) = 3 \cdot 4^{x}$ because the initial value is 3 and the $y$ -value is multiplied by 4 when the $x$ -value increases by 1, so the growth factor is 4.

$f$ and $g$ represent the number, in thousands, of social media followers of two organizations as a function of years since 2010. What does the intersection of $f$ and $g$ mean in this context?

Compare your answer:

It represents the time, in years since 2010, when the two organizations have the same number of followers.