Determining Exponential Function from Its Graph – Sahabat Smaridasa
📐 Determining Exponential Function from Its Graph
Sahabat Smaridasa — After learning about
drawing graphs of exponential functions, we continue with the topic of
Determining the Exponential Function from Its Graph.
In drawing exponential function graphs, the function is given and we are asked to draw the graph.
The reverse happens for determining the exponential function from its graph: we are given the graph
and we must determine the function. Determining the exponential function from its graph is also one of the
question types that appear in the National Examination.
Actually, for the National Examination, Determining the Exponential Function from Its Graph is not difficult
because we don't need to memorize many formulas. Instead, we simply use the SUBSTITUTION TECHNIQUE on the points
that the graph passes through for each option (multiple choice). We will try several types of multiple-choice questions.
The main skill needed here is only arithmetic proficiency.
To make it easier to study Determining the Exponential Function from Its Graph, you should master
properties of exponents for calculation purposes, the form of exponential functions, and finally
solving systems of equations. In this discussion, we broadly divide it into two types of graphs. Let's follow the discussion below.
📊 Method I: Determining Exponential Function from Its Graph
General Approach
There are generally two exponential functions that we will use as models: $ f(x) = b \cdot a^x $ and
$ f(x) = b \cdot a^x + c $. The form $ f(x) = b \cdot a^x $ is used if the exponential function graph
passes through only two points. The form $ f(x) = b \cdot a^x + c $ is used if the graph passes through
more than two points.
Important note: The exponential graphs discussed in this article are monotonic graphs,
either monotonically increasing or decreasing.
Example 1
Determine the exponential function from the following graph.
Solution:
The graph in Example 1 passes through two points: (0,1) and (1,3). So we use the model $ f(x) = b \cdot a^x $.
Substitute both points:
For $(x,y) = (0,1)$:
\[
\begin{aligned}
f(x) &= b \cdot a^x \\
1 &= b \cdot a^0 \\
1 &= b \cdot 1 \\
b &= 1
\end{aligned}
\]
So the function becomes: $ f(x) = a^x $.
For $(x,y) = (1,3)$:
\[
\begin{aligned}
f(x) &= a^x \\
3 &= a^1 \\
a &= 3
\end{aligned}
\]
So the function is: $ f(x) = 3^x $.
Answer: The exponential function from the graph is $ f(x) = 3^x $.
Example 2
Determine the exponential function from the following graph.
Solution:
The graph in Example 2 passes through two points: (1,6) and (2,12). So we use the model $ f(x) = b \cdot a^x $.
Substitute both points:
For $(x,y) = (1,6)$:
\[
\begin{aligned}
f(x) &= b \cdot a^x \\
6 &= b \cdot a^1 \\
6 &= ba \\
a &= \frac{6}{b} \quad ... \text{(i)}
\end{aligned}
\]
For $(x,y) = (2,12)$:
\[
\begin{aligned}
f(x) &= b \cdot a^x \\
12 &= b \cdot a^2 \\
12 &= ba^2 \quad ... \text{(ii)}
\end{aligned}
\]
Substitute $ a = \frac{6}{b} $ into equation (ii):
\[
\begin{aligned}
12 &= b \left(\frac{6}{b}\right)^2 \\
12 &= b \left(\frac{36}{b^2}\right) \\
12 &= \frac{36}{b} \\
b &= \frac{36}{12} = 3
\end{aligned}
\]
So $ a = \frac{6}{b} = \frac{6}{3} = 2 $.
Thus the function is: $ f(x) = b \cdot a^x = 3 \cdot 2^x $.
Answer: The exponential function from the graph is $ f(x) = 3 \cdot 2^x $.
Example 3
Determine the exponential function from the following graph.
Solution:
The graph in Example 3 passes through two points: (0,4) and (1,2). So we use the model $ f(x) = b \cdot a^x $.
Substitute both points:
For $(x,y) = (0,4)$:
\[
\begin{aligned}
f(x) &= b \cdot a^x \\
4 &= b \cdot a^0 \\
4 &= b \cdot 1 \\
b &= 4
\end{aligned}
\]
So the function becomes: $ f(x) = 4 \cdot a^x $.
For $(x,y) = (1,2)$:
\[
\begin{aligned}
f(x) &= 4 \cdot a^x \\
2 &= 4 \cdot a^1 \\
2 &= 4a \\
a &= \frac{2}{4} = \frac{1}{2}
\end{aligned}
\]
So the function is: $ f(x) = 4 \cdot \left(\frac{1}{2}\right)^x $.
Simplify:
\[
\begin{aligned}
f(x) &= 4 \cdot \left(\frac{1}{2}\right)^x \\
f(x) &= 2^2 \cdot (2^{-1})^x \\
f(x) &= 2^2 \cdot 2^{-x} \\
f(x) &= 2^{2-x}
\end{aligned}
\]
Answer: The exponential function from the graph is $ f(x) = 2^{2-x} $.
Example 4
Determine the exponential function from the following graph.
Solution:
The graph in Example 4 passes through three points: (0,4), (1,7), and (2,13). So we use the model $ f(x) = b \cdot a^x + c $.
Substitute all three points:
For $(x,y) = (0,4)$:
\[
\begin{aligned}
f(x) &= b \cdot a^x + c \\
4 &= b \cdot a^0 + c \\
4 &= b + c \quad ... \text{(i)}
\end{aligned}
\]
For $(x,y) = (1,7)$:
\[
\begin{aligned}
f(x) &= b \cdot a^x + c \\
7 &= b \cdot a^1 + c \\
7 &= ba + c \quad ... \text{(ii)}
\end{aligned}
\]
For $(x,y) = (2,13)$:
\[
\begin{aligned}
f(x) &= b \cdot a^x + c \\
13 &= b \cdot a^2 + c \\
13 &= ba^2 + c \quad ... \text{(iii)}
\end{aligned}
\]
Eliminate equations (i) and (ii):
\[
\begin{array}{cc}
ba + c = 7 & \\
b + c = 4 & - \\
\hline
ba - b = 3 &
\end{array}
\]
So we get: $ ba - b = 3 $ ... (iv).
Eliminate equations (ii) and (iii):
\[
\begin{array}{cc}
ba^2 + c = 13 & \\
ba + c = 7 & - \\
\hline
ba^2 - ba = 6 & \\
a(ba - b) = 6 &
\end{array}
\]
So we get: $ a(ba - b) = 6 $ ... (v).
From equations (iv) and (v):
\[
a(ba - b) = 6 \rightarrow a \times 3 = 6 \rightarrow a = 2
\]
From equation (iv): $ ba - b = 3 \rightarrow 2b - b = 3 \rightarrow b = 3 $.
From equation (i): $ b + c = 4 \rightarrow 3 + c = 4 \rightarrow c = 1 $.
So the function is: $ f(x) = b \cdot a^x + c = 3 \cdot 2^x + 1 $.
Answer: The exponential function from the graph is $ f(x) = 3 \cdot 2^x + 1 $.
📊 Method II: Determining Exponential Function from Its Graph (Multiple Choice)
Multiple Choice Approach
How do we determine the exponential function when the questions are in multiple-choice format (like in the National Examination)?
The best method, aside from the approach above, is to directly check each option by substituting the points that the graph passes through.
The correct function is the one that passes through all the given points.
Example 5
Observe the following function graph.
From the graph, which function represents the graph?
A) $ f(x) = 3^x + 1 $ B) $ f(x) = 2^{x-1} + 3 $
C) $ f(x) = \left(\frac{1}{2}\right)^x + \frac{7}{2} $ D) $ f(x) = {}^2\!\log x + 4 $
E) $ f(x) = {}^3\!\log (x+2) + 3 $
Solution:
Substitute the points that the graph passes through into each option. A trick for choosing points is to pick a point other than the first one, as many options will usually satisfy it. So we choose the second point: (2,5). This means when we substitute $ x = 2 $, the function value must be 5, i.e., $ f(2) = 5 $.
Answer: The function of the graph is $ f(x) = 2^{x-1} + 3 $.
Thus concludes our discussion on Determining the Exponential Function from Its Graph along with examples.
Next, please also read other related materials on exponents by following the related articles below.
