5. Exponents, Radicals, & Exponential Functions

**Exponential relationships** include any relations where the outputs increase by a **constant factor** or decrease by a constant factor for consistent changes in x.

An exponential relationship can be modeled by a function with the independent variable in the exponent, known as an **exponential function**:

\displaystyle f\left(x\right)=ab^x

\bm{a}

Leading coefficient

\bm{b}

Base where b \gt 0,\, b\neq 1

\bm{x}

Independent variable

\bm{f(x)}

Dependent variable

Consider the following equations with a=1:

y=5^x

x | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|

y | \dfrac{1}{25} | \dfrac{1}{5} | 1 | 5 | 25 |

y=2^x

x | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|

y | \dfrac{1}{4} | \dfrac{1}{2} | 1 | 2 | 4 |

For each of the functions, think about the following questions:

- What happens to y as x increases?
- Compare y=5^x and y=2^x. How are they similar? How are they different?
- What is the y-intercept for each of the functions? How does this relate to the value of a?
- Does either function have an x-intercept?
- Create a table of values for y=1^x. Does it have the same properties as y=5^x and y=2^x?

We can determine whether a function is exponential by dividing consecutive function values to see if they have a constant factor.

The base, or constant factor, is the number being multiplied repeatedly. It tells us how quickly the output values are growing or shrinking. We can find the base by dividing a term by the previous term, as shown below:

In this example, we see that the function is growing exponentially. A function grows exponentially when it increases by a constant factor.

Exponential functions change at a faster rate than linear functions. In the table below, we are adding 3 to each term, but the terms do not grow as quickly.

An exponential function can get infinitely close to an asymptote, but it can never cross it. This means that an exponential function of this form will not have an x-intercept.

Consider the table of values for the function y = 3(2)^{ x }.

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

y | \dfrac{3}{8} | \dfrac{2}{4} | \dfrac{3}{2} | 3 | 6 | 12 | 24 | 48 | 96 | 3072 |

a

Describe the behavior of the function as x increases.

Worked Solution

b

Determine the y-intercept of the function.

Worked Solution

c

State the domain of the function.

Worked Solution

d

State the range of the function.

Worked Solution

A population of bacteria can be modeled with the equation p(t)=100\left(2\right)^t, where p(t) is the population after t days. This graph shows the population over time.

a

Identify and interpret the p-intercept.

Worked Solution

b

Estimate and interpret when p(t)=1600.

Worked Solution

c

Identify and interpret the domain and range in this context.

Worked Solution

Idea summary

The base of the exponent is the constant factor, or the number being multiplied repeatedly. We can find it by dividing one output by the previous output.

All exponential functions of the form y=ab^x have the following features in common:

- The domain is -\infty \lt x \lt \infty.
- The range is y>0.
- If there is a context, the domain and range may be different based on the realistic constraints.
- The y-intercept is at (0,\,a).
- There is a horizontal asymptote at y=0.