Determine the initial value of an exponential function

For an exponential function written in the form $f(x)=ab^x$ , the initial value is $f(0)=a$ , because $b^0=1$ ; it is the $y$ -intercept and represents the quantity at the starting input in a context. The initial value can be identified from an equation, table, graph, or situation, while distinguishing it from $b$ , the multiplicative growth or decay factor; shifted forms, parameter estimation, and more generalized exponential models are beyond this scope.

Detailed Explanation: Determine the initial value of an exponential function

For an exponential function written as

f(x)=ab^x,

the initial value is the value of the function when $x=0$ :

f(0)=a.

This works because any nonzero number raised to the zero power equals $1$ :

b^0=1.

So,

f(0)=a\cdot b^0=a\cdot 1=a.

The initial value is also the $y$ -intercept. It represents the quantity at the starting input, $x=0$ . Do not confuse it with $b$ , which is the growth or decay factor.

Example

Suppose the number of bacteria in a sample is modeled by

P(t)=500(1.2)^t,

where $t$ is the number of hours.

Step 1: Identify the form.

Compare the equation with

f(x)=ab^x.

In this equation:

$a=500$
$b=1.2$

Step 2: Identify the initial value.

The initial value is $a$ , so

P(0)=500.

You can also verify it by substituting $t=0$ :

P(0)=500(1.2)^0

P(0)=500(1)=500.

Therefore, the initial number of bacteria is $500$ . The number $1.2$ is the growth factor, not the initial value.

Learn by doing: Determine the initial value of an exponential function

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Exponential Function Growth (Discrete) - Equation to Scenario

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