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All courses Math III · F-IF.7.e 26 of 55
Graph exponential, logarithmic, and trigonometric functions with key features.

Graph exponential growth from complete feature fields

Problem
For the exponential growth function f(x)=3·2ˣ, enter a,b,h,k, factor classification, y-intercept, horizontal asymptote, domain/range, monotonicity, and repeated exact anchor points.
Prompt coordinate grid scaled for the stated exponential equation; the curve, anchors, and asymptote are withheld. Open full size
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Hint

Identify a,b,h,k and evaluate f(0) separately from the shifted anchor.

For a·bˣ⁻h+k with a>0,b>1: asymptote y=k, range (k,∞), and parent anchor (0,1) maps to (h,a+k).

Solution walkthrough

01

Identify the form of the function

\[f(x)~=~3*2^x\]

This matches the exponential growth form a(b)ˣ with a=3 and b=2. In that form, a helps you find the y-intercept and b is the growth factor.

02

Find the y-intercept

\[f(0)~=~3*2^0~=~3*1~=~3\]

The y-intercept happens when x=0. Since 2⁰=1, the graph crosses the y-axis at (0,3), so the y-intercept is 3.

03

Find the asymptote and growth factor

\[\begin{aligned} \text{horizontal}~\text{asymptote}:~y~=~0 \\ \text{growth}~\text{factor}:~2 \end{aligned}\]

There is no +k outside the exponential term, so the graph approaches y=0. The base is 2, so each time x increases by 1, the output is multiplied by 2.

04

State the graphing features

\[\text{parameters}=a=3,b=2,h=0,k=0;~\text{factor}~\text{classification}=\text{growth},b>1;~y-\text{intercept}=(0,3);~\text{horizontal}~\text{asymptote}=y=0;~\text{domain}=(-∞,∞);~\text{range}=(0,∞);~\text{monotonicity}=\text{increasing};~\text{anchor}~\text{rows}=(-1,3/2);(0,3);(1,6)\]

These are the key features you would use to sketch the graph: it starts at 3 on the y-axis, stays above the x-axis, and doubles each step to the right.

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Another way

  1. You can verify the growth factor by comparing points: f(0)=3 and f(1)=6, so the output doubles from 3 to 6. That confirms a growth factor of 2.

!

Common mistake

A common mistake is calling 3 the growth factor because it is the number in front. In a function of the form a(b)ˣ, the front number a gives the y-intercept, while the base b gives the growth factor.

Answer exponential graph with the exact curve, horizontal asymptote, anchor points, domain, range, and monotonic direction marked.
The completed exponential graph verifies the requested exact features.