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All courses Algebra II · F-IF.7.e 33 of 55
Graph exponential, logarithmic, and trigonometric functions with key features.

Graph exponential growth from complete feature fields

Problem
Analyze \(f(x)~=~3~\cdot~2^x\). State \(a,~b,~h,~\text{and}~k\); classify growth or decay; give the intercepts, asymptote, domain, range, and monotonic behavior; and compute exact anchors at \(x~=~-1\), \(0\), and \(1\).
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

Match the exponential form

\[f(x)=a*b^{x-h}+k=3*2^{x-0}+0\]

Comparing the given function with the transformed exponential form gives a=3, b=2, h=0, and k=0. The coefficient 3 is a vertical scale, not a horizontal shift.

02

Classify and find the main features

\[b=2>1~->~\text{growth};~f(0)=3*2^0=3;~y=k=0\]

A base greater than 1 produces growth. Evaluating at x=0 gives the y-intercept (0,3), while the outside shift k gives the horizontal asymptote y=0. Positive exponential outputs make the range y>0; the domain is all real numbers, and the function is increasing.

03

Compute exact anchors

\[f(-1)=3*2^{-1}=3/2;~f(0)=3;~f(1)=3*2=6\]

Substitute each requested x-value into the original function. These points also check the growth factor: each step right doubles the output.

04

State the complete analysis

\[\text{Parameters}:~a~=~3,~b~=~2,~h~=~0,~k~=~0.~\text{Class}:~\text{growth}~\text{because}~b~>~1.~\text{Features}:~y-\text{intercept}~(0,~3);~\text{asymptote}~y~=~0;~\text{domain}~\text{all}~\text{real}~\text{numbers};~\text{range}~y~>~0;~\text{increasing}.~\text{Anchors}:~(-1,~3/2),~(0,~3),~(1,~6).\]

All requested parameters, graph features, and exact anchors agree with the function, so this is choice A.

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Another way

  1. Start from the parent anchors (-1,1/2), (0,1), and (1,2), then multiply every y-coordinate by 3.

!

Common mistake

Do not call x=0 a vertical asymptote or use 3 as a shift. For a*bˣ⁻h+k, the asymptote is the horizontal line y=k, and the y-intercept must be found by evaluating the whole function at x=0.

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.

Solution walkthrough video