Skill: 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.