Question

which graph represents a function with a growth factor of 5?

Explanation:

Step1: Recall exponential growth form

An exponential growth function has the form $f(x)=a(b)^x$, where $b$ is the growth factor ($b>1$). For a growth factor of 5, the function is $f(x)=a(5)^x$.

Step2: Test integer x-values

For $a=1$, calculate points:

• When $x=0$, $f(0)=5^0=1$
• When $x=1$, $f(1)=5^1=5$
• When $x=2$, $f(2)=5^2=25$

This means the graph rises very steeply: from (0,1) to (1,5) is a vertical jump of 4 units, which is much steeper than graphs with smaller growth factors (like growth factor 2, which goes from (0,1) to (1,2)).

Step3: Match steepness to graphs

Among the options, the third graph (left to right) shows the steepest upward curve that matches the rapid growth of $f(x)=5^x$.

Answer:

The third graph (left-to-right, the one with the steepest vertical rise from the x-axis)