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Math Lesson 15.5.4 - How the Value of the Base a Affects the Graph Shape?
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Welcome to our Math lesson on How the Value of the Base a Affects the Graph Shape?, this is the fourth 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.
How the Value of the Base a Affects the Graph Shape?
Now, let's see what happens to the exponential graph for different values of the base a. For this, we will take the graph of y(x) = 2x as a reference. Thus, we will plot three graphs in the same coordinates system: y = 2x, y = (1/4)x and y = 3x, as shown below.
From the figure, we draw the following conclusions about exponential graphs:
- For a > 1, when the base a increases, the graph becomes 'narrower'.
- For a < 1, the graph flips horizontally in respect to the corresponding exponential graph with a reciprocal base.
Example 4
The following figure shows four graphs A, B, C and D, where the black one (graph A) is y = ax.
What can you say about the base a of the other graphs B, C and D compared to that of graph A if all graphs have the same kind of formula? Order them from the smallest to the biggest.
Solution 4
From theory, it is known that when the graph becomes narrower, the base a increases. Hence, we say aC > aA.
On the other hand, when the exponential graph becomes wider, the base a decreases but if the direction of the graph is the same, then its value is still greater than 1. Hence, aB < aA.
When the exponential graph changes direction, i.e. when it looks horizontally flipped, the value of the base a is between 0 and 1. Hence, aB < 1.
Therefore, the order of the coefficients a for the four exponential graphs shown in the figure is
You have reached the end of Math lesson 15.5.4 How the Value of the Base a Affects the Graph Shape?. There are 8 lessons in this physics tutorial covering Exponential Graphs, you can access all the lessons from this tutorial below.
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