Exponentials are very similar to polynomials and are easily confused. Check out the next two examples.

The polynomial you should be familiar with. Notice that the difference between a polynomial and an exponential is where the variable is, in the base or in the exponent. In the base, we have a polynomial, in the exponent, we have an exponential function.

In general, an exponential function is written \(f(x) = Ca^x\) where a is a positive real number (\(a \gt 0\)) and \(a \neq 1 \) and C is a real number.
a is called the growth factor
C is called the initial value since \(f(0) = Ca^0 = C\)

Remember these laws of exponents? These also hold for exponential functions.

Also remember that \((a+b)^s \neq a^s + b^s\). An example of this is \((a+b)^2 = a^2 +2ab + b^2 \neq a^2 + b^2\).
Before we go on, here is a good video that shows several examples so that you can get a feel for exponential functions.

Notice we are calling \(g(x)=a^x\) a function. The use of the word 'function' is deliberate since exponentials pass the vertical line test. Let's look at a few exponential functions. We will draw them on the same graph so that we can compare them.

In plot 1 below, notice that all of the exponential functions have the same general shape and that they all go through the point \((0,1)\). This makes sense, since \(2^0=3^0=4^0=1\). This shape holds for exponential functions \(y=a^x\) where \(a>1\).

Now take a careful look at plot 2. Notice that the exponential functions are still in the form \(y=a^x\) but in these cases \(0 < a < 1\). Said another way, \(a > 1\) and \(y=(1/a)^x\). And said still another way, \(a > 1\) and \(y=a^{-x}\).

Again, all plots go through the point \((0,1)\) and they all have the same general shape.

When comparing the two plots, notice that they are flipped with respect to the y-axis. These are the kind of things you need to look at and think about in calculus when you are given a graph.

plot 1: \(\color{red}{y=2^x}\); \(\color{blue}{y=3^x}\); \(\color{green}{y=4^x}\)

plot 2: \(\color{red}{y=(1/2)^x}\); \(\color{blue}{y=(1/3)^x}\); \(\color{green}{y=(1/4)^x}\)

Okay, let's watch a video clip that explains this in more detail.

Time for some practice problems.

Practice

Unless otherwise instructed, graph these exponential functions.

• Practice 2987
• Solution

Sketch the graph of \( y=2^x \) and find the domain and range.

Problem Statement

Sketch the graph of \( y=2^x \) and find the domain and range.

Solution

2987 video

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• Practice 2988
• Solution

Sketch the graph of \(y=(1/3)^x \) and find the domain and range.

Problem Statement

Sketch the graph of \(y=(1/3)^x \) and find the domain and range.

Solution

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• Practice 2989
• Solution

Sketch the graph of \( y = 3^{x-2} + 1 \) and find the domain and range.

Problem Statement

Sketch the graph of \( y = 3^{x-2} + 1 \) and find the domain and range.

Solution

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• Practice 2990
• Solution

Sketch the graph of \( y= 5 - 2^{3-x} \) and find the domain and range.

Problem Statement

Sketch the graph of \( y= 5 - 2^{3-x} \) and find the domain and range.

Solution

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• Practice 1693
• Solution

Unless otherwise instructed, graph the exponential function \(y=2^x+1\).

Problem Statement

Unless otherwise instructed, graph the exponential function \(y=2^x+1\).

Solution

1693 video

video by PatrickJMT

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• Practice 1694
• Solution

Unless otherwise instructed, graph the exponential function \(y=3^{-x}-2\).

Problem Statement

Unless otherwise instructed, graph the exponential function \(y=3^{-x}-2\).

Solution

1694 video

video by PatrickJMT

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• Practice 1695
• Solution

Unless otherwise instructed, graph the exponential function \(y=3(1/2)^x\).

Problem Statement

Unless otherwise instructed, graph the exponential function \(y=3(1/2)^x\).

Solution

1695 video

video by PatrickJMT

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• Practice 1696
• Solution

Unless otherwise instructed, graph the exponential function \(y=-3^x-1\).

Problem Statement

Unless otherwise instructed, graph the exponential function \(y=-3^x-1\).

Solution

1696 video

video by PatrickJMT

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• Practice 1697
• Solution

Determine the equation of this transformed exponential function.

Problem Statement

Determine the equation of this transformed exponential function.

Solution

1697 video

video by MIP4U

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• Practice 1690
• Solution

Unless otherwise instructed, graph the exponential function \(y=5^x\).

Problem Statement

Unless otherwise instructed, graph the exponential function \(y=5^x\).

Solution

1690 video

video by Khan Academy

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There is a special exponential function that is used extensively in calculus, \(f(x)=e^x\). The number \(e \approx 2.7183\) is a special irrational number. (You need calculus to understand where it comes from.) Since \(e > 1\), the graph looks like plot 1 above.

In order to get a feel for working with exponential functions, you may be asked to solve exponential functions. The first technique does not require the knowledge of logarithms. This next video contains an explanation how to do this followed by lots of examples.

Here are some videos to help you review rules of exponents. These videos also contain plenty of examples.

Exponentials are very closely related to logarithms. So, after working the practice problems on this page, your next logical step is logarithms. After that, we will discuss how to use exponentials and logarithms together and then solve some application problems.