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Concept

Exponential Function

An exponential function is a nonlinear function that can be written in the following form, where a0, b>0, and b1. As the independent variable x changes by a constant amount, the dependent variable y multiplied by a constant factor. Therefore, consecutive y-values form a constant ratio.

Here, the coefficient a is the y-intercept, which is sometimes referred to as the initial value. The base b can be interpreted as the constant factor. The graph of an exponential function depends on the values of a and b.
Graph of y=a*b^x for a-values less than and greater than 0, and for b-values less than and greater than 1.
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Why

Why a0?

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If the coefficient a is 0, the function becomes a horizontal line.
This is a line along the x-axis and, therefore, is a linear relation. This means that if a=0, then the function is not exponential.
Graph of y=a*2^x where the value of 'a' can be changed from -2.5 to 2.5
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Further explanation about a difficult or interesting topic.

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Why

Why and b1?

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If the base b is negative, the function gives undefined results for certain x-values. For example, since a negative value for b would yield non-real values for Hence, a condition on b is needed.
However, if b=0 or b=1, the function becomes a horizontal line.
Therefore, b cannot be equal to 0 nor 1.
Graph of y=2*b^x where the value of 'b' can be changed from 0.1 to 2
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