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All courses Math I · F-IF.7.e 26 of 59
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.

Graph an exponential growth function

Problem
Graph f(x) = 3·2ˣ on the grid. State the initial value, growth factor and percent, verified points and ratio, horizontal asymptote, domain, and range.
Blank coordinate grid tailored to plotting f(x)=3·2ˣ, without a completed curve or derived features. Open full size
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Hint

Compare \(f(x)=3(2)^x\) to \(a(b)^x\) so you can identify the coefficient and the base before choosing points.

For exponential functions in the form \(a(b)^x\), the point at \(x=0\) is \((0,a)\) because \(b^0=1\). Then multiply by the growth factor each time x increases by 1.

Solution walkthrough

01

Identify the initial value and growth factor

\[f(x)~=~3(2)^x~->~\text{initial}~\text{value}~=~3,~\text{growth}~\text{factor}~=~2\]

In an exponential function \(a(b)^x\), the coefficient \(a\) is the starting value when \(x=0\), and the base \(b\) is the growth factor.

02

Find the first point

\[f(0)~=~3(2)^0~=~3(1)~=~3~->~(0,3)\]

Any nonzero base to the zero power is 1, so the graph starts at \((0,3)\).

03

Find two more points

\[\begin{aligned} f(1)~=~3(2)^1~=~6~->~(1,6) \\ f(2)~=~3(2)^2~=~12~->~(2,12) \end{aligned}\]

As x increases by 1, the output is multiplied by 2. Starting from 3, the next values are 6 and 12.

04

State the graphing information

\[\text{initial}~=~f(0)=3;~\text{growth}~\text{factor}~=~2;~\text{growth}~\text{percent}~=~100\%~\text{per}~+1~\text{input};~\text{points}~=~(0,3),(1,6),(2,12);~\text{ratio}~\text{check}~=~6/3=12/6=2;~\text{asymptote}~=~y=0;~\text{domain}~=~(-∞,∞);~\text{range}~=~(0,∞)\]

These values give the key information to sketch the graph: the function starts at 3 and doubles each step to the right.

+

Another way

  1. You can build the points by repeated multiplication instead of evaluating each exponent separately: start at 3, then multiply by 2 to get 6, then multiply by 2 again to get 12.

!

Common mistake

A common mistake is to call 3 the growth factor because it is in front. In \(a(b)^x\), the front number is the initial value, while the base is the growth factor.

Actual graph of f(x)=3·2ˣ with verified points, intercept, asymptote, and end behavior.
initial 3; factor 2; points (0,3),(1,6),(2,12); asymptote y=0