Graphing Exponential Decay Functions: f(x) = 12(0.3)^x

Published on May 21, 2026

Functions

Learn how to graph an exponential decay function f(x) = 12(0.3)^x and analyze its key features including y-intercept, asymptote, range, and end behavior.

Question

Graph the function f (x) = 12(0.3)x, and use the graph to answer the following questions. Part A: Describe the graph of f (x) using two of the following key features: y-intercept, asymptote, interval where the function is increasing or decreasing, and interval where the function is positive or negative. Part B: What is the range of f (x)? Use the graph to explain the answer. Part C: What is the end behavior of f (x)? Use the graph to explain the answer.

Answer

To solve this problem, we first analyze the function

. This is an exponential decay function because the base

is between 0 and 1.

Analysis of the Function

y-intercept: To find the y-intercept, set :
. The y-intercept is .
Asymptote: As increases, approaches 0. Therefore, the horizontal asymptote is the x-axis, or .
Behavior: Since the base is , the function is strictly decreasing over its entire domain .
Positivity: Since and is always positive for any real , the function is positive for all .

Part A: Key Features

We can describe the graph using the following two features:

y-intercept: The graph crosses the y-axis at .
Interval of decrease: The function is decreasing over the interval because as increases, the value of consistently drops.

(Alternatively, one could mention the horizontal asymptote at or that the function is positive on ).

Part B: Range

The range of

, or

Explanation using the graph:
Looking at the graph, the curve stays entirely above the x-axis. While the values of

get closer and closer to 0 as

moves toward positive infinity, the graph never actually touches or crosses the x-axis (

). As

moves toward negative infinity, the graph rises indefinitely toward positive infinity. Therefore, the output values are all real numbers greater than 0.

Part C: End Behavior

The end behavior describes what happens to

approaches positive and negative infinity:

As : As the graph moves to the right, the curve flattens out and approaches the horizontal asymptote .
As : As the graph moves to the left, the curve rises steeply toward positive infinity.

Explanation using the graph:
The graph shows a steep climb on the left side (negative x-axis) and a leveling off toward the x-axis on the right side (positive x-axis).

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