Question

The graph of an exponential function is given. Select the function for each graph from the following options: ﻿f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.f(x) = 3^x, \quad g(x) = $$3^{x-1}$$, \quad h(x) = 3^x - 1, \
f(x) = -3^x, \quad G(x) = $$3^{-x}$$, \quad H(x) = $$-3^{-x}.
f(x)=3x$$,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.

Accepted Answer

Step 1: Identify the horizontal asymptote of the graph. The graph shows a horizontal asymptote at $$y = -1$$, which means the function approaches $$-1 as$$ $$x$$ goes to negative infinity. Step 2: Recall that the basic exponential function $$3^x$$ has a horizontal asymptote at $$y = 0. To $$shift this asymptote to $$y = -1$$, the function must be vertically shifted down by 1 unit. This suggests the function has the form $$3^x - 1. $$Step 3: Check the point given on the graph, which is $$(1, 5). $$Substitute $$x = 1$$ into the candidate function $$3^x - 1 to $$verify: $$3^1 - 1 = 3 - 1 = 2$, which does not match the point (1$$, 5)$$. Step 4: Consider the function 3$$^{x-1}$$, which shifts the graph horizontally to the right by 1 unit. The asymptote remains at y = 0$, so this does not match the asymptote at y = -1$. Step 5: Since the asymptote is at y = -1$ and the graph passes through (1$$, 5)$$, the function must be f(x) = 3^x - 1$. The discrepancy in step 3 suggests rechecking the point or considering the vertical shift as the key feature to identify the function.

The graph of an exponential function is given. Select the function for each graph from the following options:
$f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \\ f(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.$

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Key Concepts

Exponential Functions and Their Graphs

Transformations of Exponential Functions

Horizontal Asymptotes in Exponential Functions