Graph of Exponential and Logarithmic Functions – Sahabat Smaridasa
📈 Graph of Exponential and Logarithmic Functions
Sahabat Smaridasa — In this article, we will discuss the
graph of exponential and logarithmic functions.
The graph of an exponential function is a monotonic graph, either monotonically increasing or decreasing.
However, in this article on Graph of Exponential and Logarithmic Functions, we will only discuss the exponential function graph.
As for the graph of logarithmic functions, we have previously shared it in an article entitled
"logarithmic functions". Please visit that article to learn about logarithmic function graphs.
Drawing the Graph of an Exponential Function is not very difficult. The simplest form of an exponential function
is $ f(x) = a^x $. Please also read the material on
"exponential functions" to make it easier to study and
create/draw the graph of an exponential function. The main factor that determines the shape of the exponential function graph
is the value of $ a $, which is called the base (please read:
General Form of Exponents or Powers). If $ a > 1 $, the graph is generally monotonically increasing, and if $ 0 < a < 1 $,
the graph is monotonically decreasing.
📊 Graph of Exponential Function $ f(x) = a^x $
Graph of $ f(x) = a^x $
For $ a > 1 $: The graph intersects the Y-axis at $ y = 1 $ and is monotonically increasing. For $ 0 < a < 1 $: The graph intersects the Y-axis at $ y = 1 $ and is monotonically decreasing.
Note: We can take several points $(x, y)$ that satisfy the exponential function by substituting chosen values of $ x $
first, then after substitution we will get the value of $ y $. These points will help us in drawing the graph more easily.
Example 1
Draw the graph of the following exponential functions:
a) $ f(x) = 2^x $ b) $ f(x) = 5^x $ c) $ f(x) = 9^x $
d) $ f(x) = \left(\frac{1}{2}\right)^x $ e) $ f(x) = \left(\frac{1}{5}\right)^x $ f) $ f(x) = \left(\frac{1}{9}\right)^x $
Solution:
For $ f(x) = 2^x, \, f(x) = 5^x, \, f(x) = 9^x $, the base is greater than 1, so the graph is monotonically increasing as shown below.
For $ f(x) = \left(\frac{1}{2}\right)^x, \, f(x) = \left(\frac{1}{5}\right)^x, \, f(x) = \left(\frac{1}{9}\right)^x $, the base is between 0 and 1, so the graph is monotonically decreasing as shown below.
Note: The graph of $ f(x) = \left(\frac{1}{a}\right)^x $ can be obtained by reflecting the graph of $ f(x) = a^x $ across the Y-axis, and vice versa.
📊 Graph of Exponential Function $ f(x) = b \cdot a^x $
Graph of $ f(x) = b \cdot a^x $
For $ a > 1 $: The graph intersects the Y-axis at $ y = b $ and is monotonically increasing. For $ 0 < a < 1 $: The graph intersects the Y-axis at $ y = b $ and is monotonically decreasing.
Example 2
Draw the graph of the exponential functions $ f(x) = 2 \cdot 5^x $ and $ f(x) = 2 \cdot \left(\frac{1}{5}\right)^x $.
Solution:
📊 Graph of Exponential Function $ f(x) = b \cdot a^x + c $
Graph of $ f(x) = b \cdot a^x + c $
For $ a > 1 $: The graph intersects the Y-axis at $ y = b + c $ and is monotonically increasing. For $ 0 < a < 1 $: The graph intersects the Y-axis at $ y = b + c $ and is monotonically decreasing.
Example 3
Draw the graph of the following exponential functions:
a) $ f(x) = 2 \cdot 3^x + 1 $ b) $ f(x) = 2 \cdot 3^x - 3 $
c) $ f(x) = 2 \cdot \left(\frac{1}{3}\right)^x + 1 $ d) $ f(x) = 2 \cdot \left(\frac{1}{3}\right)^x - 3 $
Solution:
For (a) and (c): $ b = 2 $ and $ c = 1 $, so the Y-intercept is $ y = 2 + 1 = 3 $.
For (b) and (d): $ b = 2 $ and $ c = -3 $, so the Y-intercept is $ y = 2 - 3 = -1 $.
Graphs (a) and (b) are monotonically increasing:
Graphs (c) and (d) are monotonically decreasing:
📊 Graph of Negative Exponential Functions
Negative Exponential Functions
The graphs of $ f(x) = -a^x $, $ f(x) = -b \cdot a^x $, and $ f(x) = -(b \cdot a^x + c) $ are obtained by reflecting
the graphs of $ f(x) = a^x $, $ f(x) = b \cdot a^x $, and $ f(x) = b \cdot a^x + c $ across the X-axis.
Example 4
Draw the graph of the following exponential functions:
a) $ f(x) = -2 \cdot 3^x $ b) $ f(x) = -2 \cdot 3^x + 3 $
Solution:
a) The graph of $ f(x) = -2 \cdot 3^x $ is obtained by reflecting the graph of $ f(x) = 2 \cdot 3^x $ across the X-axis.
b) The graph of $ f(x) = -2 \cdot 3^x + 3 = -(2 \cdot 3^x - 3) $ is obtained by reflecting the graph of $ f(x) = 2 \cdot 3^x - 3 $ across the X-axis.
Thus concludes our discussion on the Graph of Exponential and Logarithmic Functions along with examples.
Next, please also read other related materials such as
determining the exponential function from its graph.
May this material be beneficial. Thank you.
