Math Lesson 15.5.6 - Finding the Formula of the Exponential Function from its Graph

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Welcome to our Math lesson on Finding the Formula of the Exponential Function from its Graph, this is the sixth lesson of our suite of math lessons covering the topic of Exponential Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Finding the Formula of the Exponential Function from its Graph

We can find the formula of an exponential function from its graph. For this, we have to consider some distinct points, as we must find four different values that correspond to the two coefficients k and m, the base a and the constant t. Let's see an example to clarify this point.

Example 5

Find the function indicated by the graph below.

$Math Tutorials: Exponential Graphs Example$

Solution 5

First, let's assume the function has the general form

y(x) = k ∙ a^mx + t

We exclude immediately the constant t, as the asymptote is y = t = 0. This means the function shown in the graph has the general form

y(x) = k ∙ a^mx

We can see in the graph that for x = 0, we have y = 2. Hence, we have

2 = k ∙ a^{m ∙ 0}
2 = k ∙ a⁰
2 = k ∙ 1

Thus,

k = 2/1 = 2

Therefore, we have the general form of the required function

y(x) = 2 ∙ a^mx

Now, we need at least the coordinates of two other points in order to find the rest of the unknown coefficients. From the figure, you can see that (2, 6) and (4, 18) are points on the graph. We have

6 = 2 ∙ a^{m ∙ 2}

and

18 = 2 ∙ a^{m ∙ 4}

a^2m = 3

and

a^4m = 9

Now, given that the second equation can be written as

(a^2m)² = 3²
a²m = 3

i.e. the two equations are identities, we focus on one of them only, where the base a obviously is is 3. Thus, since

3²m = 3¹

we have

2m = 1
m = 1/2

Hence, the function shown in the graph is

y(x) = 2 ∙ 3^{1/2 x}

You have reached the end of Math lesson 15.5.6 Finding the Formula of the Exponential Function from its Graph. There are 8 lessons in this physics tutorial covering Exponential Graphs, you can access all the lessons from this tutorial below.

More Exponential Graphs Lessons and Learning Resources

Types of Graphs Learning Material
Tutorial ID	Math Tutorial Title	Revision Notes	Revision Questions
15.5	Exponential Graphs
Lesson ID	Math Lesson Title	Lesson	Video Lesson
15.5.1	What are Exponential Functions?
15.5.2	Exponential Function's Graph. Asymptotes of Exponential Graphs
15.5.3	Plotting Exponential Graphs
15.5.4	How the Value of the Base a Affects the Graph Shape?
15.5.5	How do the Values of the Other Coefficients Affect the Graph?
15.5.6	Finding the Formula of the Exponential Function from its Graph
15.5.7	Exponential Graphs with Negative Base
15.5.8	Exponential Graphs that Involve Euler's Number

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Types of Graphs Math tutorial: Exponential Graphs. Read the Exponential Graphs math tutorial and build your math knowledge of Types of Graphs
Types of Graphs Revision Notes: Exponential Graphs. Print the notes so you can revise the key points covered in the math tutorial for Exponential Graphs
Types of Graphs Practice Questions: Exponential Graphs. Test and improve your knowledge of Exponential Graphs with example questins and answers
Check your calculations for Types of Graphs questions with our excellent Types of Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Types of Graphs Calculators by iCalculator™ below.
Continuing learning types of graphs - read our next math tutorial: Circle Graphs

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