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All courses Algebra I · F-IF.7.e 26 of 76
Graph exponential functions and identify intercepts and end behavior; preview logarithmic and trigonometric graph features.

Graph an exponential growth function

Problem
Graph \(f(x)~=~3\cdot~2^{x}\) 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

Decode the exponential parameters

\[f(x)=a(b)^x;~a=3;~b=2\]

The coefficient 3 is the output at x=0, and the base 2 is the multiplicative factor for each increase of 1 in x. A factor of 2 means a 100% increase because 2-1=1.

02

Calculate and check graph points

\[f(0)=3;~f(1)=6;~f(2)=12;~6/3=12/6=2\]

The points (0,3), (1,6), and (2,12) lie on the rule, and their equal consecutive ratios verify exponential growth by factor 2.

03

Determine asymptote, domain, and range

\[\text{asymptote}~y=0;~\text{domain}=(-\infty,\infty);~\text{range}=(0,\infty)\]

The rule accepts every real input and always gives a positive output. With no vertical shift, the curve approaches but never reaches y=0.

04

State the complete graph information

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

The inspected grid spans x=-2 to 3 and displays y=0 inside its vertical window; its answer asset agrees with the calculated points, increasing curve, and asymptote.

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Another way

  1. Start at (0,3) and repeatedly multiply outputs by 2 for each one-unit move right, then check the same ratio backward for negative inputs.

!

Common mistake

Do not swap coefficient and base. Three is the initial value, while two is the growth factor; a factor of two means 100% growth, not 2% growth.

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

Solution walkthrough video