Graphing Exponents

What Are Exponential Functions?

An exponential function is a mathematical function in which the variable appears in the exponent. These functions are used to model situations involving rapid growth or decline, like the spread of a virus or the depreciation of a car's value. While you may be used to functions like linear equations (e.g., $y=2x+3$) where the variable $x$ is the base, in exponential functions, $x$ is the power.

The general form of an exponential function is:

Let's break down this formula:

• $y$ is the final amount.
• $a$ is the initial amount or the starting value. It's also the $y$-intercept of the graph, which is the point where the graph crosses the vertical $y$-axis. The value of $a$ cannot be zero.
• $b$ is the base, which represents the growth or decay factor. The base $b$ must be a positive number and cannot be equal to $1$.
• $x$ is the exponent, which is our independent variable. It often represents time or the number of intervals.

The key feature that makes these functions unique is that they change by a constant multiplicative factor. In a linear function, you add or subtract the same amount for each step in $x$. In an exponential function, you multiply by the same amount (the base $b$) for each step in $x$, leading to much faster changes.

How Do You Graph the Basic Exponential Function, $y=b^x$?

The simplest way to start graphing is by looking at the parent function, where the initial value $a$ is $1$. This simplifies our equation to $y=b^x$. The best method to graph any new function is to create a table of values. We'll pick a few $x$-values—some negative, zero, and some positive—and calculate the corresponding $y$-values.

Let's walk through an example of exponential growth, where the base $b$ is greater than $1$.

Notice that as $x$ gets more negative, the $y$-values get closer and closer to zero but never actually reach it. This leads to a key feature called an asymptote, which we'll discuss later.

What's the Difference Between Exponential Growth and Decay?

The behavior of an exponential graph is entirely determined by its base, $b$. This single value tells you whether you're looking at a graph that's rapidly increasing (growth) or one that's rapidly decreasing (decay).

• Exponential Growth: This occurs when the base $b>1$. For every one-unit increase in $x$, the $y$-value is multiplied by a factor greater than one, causing it to increase. The graph rises from left to right. Think of a bank account with compound interest.
• Exponential Decay: This occurs when $0<b<1$. The base is a fraction or decimal between 0 and 1. For every one-unit increase in $x$, the $y$-value is multiplied by this fraction, causing it to decrease. The graph falls from left to right. Think of the value of a new phone a year after you buy it.

Let's graph an example of exponential decay to see the difference.

How Does the 'a' Value Affect the Graph?

Now let's bring back the $a$ value from our general form $y=a\cdot b^x$. This coefficient plays two important roles: it determines the $y$-intercept and it vertically stretches or compresses the graph.

The $y$-intercept: The $y$-intercept is the point where $x=0$. If we plug $x=0$ into the equation, we get $y=a\cdot b^0$. Since any positive number raised to the power of 0 is 1, this simplifies to $y=a\cdot 1$, or just $y=a$. So, the $y$-intercept is always at the point $(0,a)$.

Vertical Stretch/Compression: The value of $a$ also acts as a scaling factor. If $|a|>1$, the graph will be stretched vertically, making it appear steeper. If $0<|a|<1$, the graph will be compressed vertically, making it appear flatter. If $a$ is negative, the entire graph is reflected across the $x$-axis.

What Is a Horizontal Asymptote in Exponential Graphs?

A horizontal asymptote is a horizontal line that the graph of a function approaches as $x$ approaches positive or negative infinity. For exponential functions in the form $y=a\cdot b^x$, the graph gets incredibly close to the $x$-axis but never actually touches or crosses it. Therefore, the $x$-axis itself serves as the horizontal asymptote.

Why does this happen? Let's consider the decay function $y=(\frac{1}{2}{)}^x$. As $x$ gets larger (e.g., $10,100,1000$), the value of $y$ becomes a tiny fraction ($\frac{1}{1024}$, etc.), getting closer and closer to $0$. However, you can never raise a positive base to a power and get an answer of zero. The result will always be a small positive number.

Similarly, for the growth function $y=2^x$, as $x$ becomes more negative (e.g., $-10,-100$), the value of $y$ becomes a tiny fraction ($\frac{1}{1024}$, etc.), again approaching $0$ from above. The asymptote acts as a boundary that the function's curve will never cross.

Note: If the function is vertically shifted, such as $y=a\cdot b^x+k$, the horizontal asymptote will also shift to the line $y=k$.

What Are the Key Features of an Exponential Graph?

When you analyze or sketch an exponential graph of the form $y=a\cdot b^x$ (with $a>0$), you should always identify these five key characteristics:

1. Y-Intercept: This is the point where the graph crosses the $y$-axis. It is always located at $(0,a)$. This is your starting point for graphing.
2. Horizontal Asymptote: This is the line the graph approaches but never touches. For the basic form, it is always the line $y=0$ (the $x$-axis).
3. Domain: The domain is the set of all possible $x$-values. For any exponential function, you can plug in any real number for $x$, from negative infinity to positive infinity. So, the domain is always $(-\mathrm{\infty },\mathrm{\infty })$.
4. Range: The range is the set of all possible $y$-values. Since the graph never touches or goes below the asymptote at $y=0$ (for $a>0$), the $y$-values are always positive. The range is $(0,\mathrm{\infty })$. If $a$ were negative, the graph would be reflected, and the range would be $(-\mathrm{\infty },0)$.
5. Growth or Decay: The graph will either be continuously increasing (growth, when $b>1$) or continuously decreasing (decay, when $0<b<1$). It will never do both.

What Common Mistakes Should I Avoid?

Graphing exponents can be tricky at first. Here are some common pitfalls to watch out for:

• Incorrect Order of Operations: A very common mistake is to multiply $a$ and $b$ before applying the exponent. In $y=3\cdot 2^x$, you must calculate $2^x$ first, then multiply by $3$. $3\cdot 2^x$ is NOT the same as $6^x$.
• Connecting Points with a Straight Line: Exponential functions produce curves, not straight lines. After plotting your points, connect them with a smooth, continuous curve that shows the rapid change in steepness.
• Crossing the Asymptote: Students sometimes mistakenly draw the curve crossing or dipping below the horizontal asymptote. Remember, the graph gets infinitely close to this line but never touches it.
• Confusing Growth and Decay: Double-check your base $b$. If $b$ is greater than 1 (like $5$ or $1.02$), it's growth. If $b$ is between 0 and 1 (like $0.5$ or $\frac{1}{3}$), it's decay.
• Mixing up $y=2^x$ and $y=x^2$: These look similar but are vastly different. In $y=2^x$, the variable is in the exponent (exponential function). In $y=x^2$, the variable is the base (a quadratic function, which graphs as a parabola).

Quick Reference: Graphing $y=a\cdot b^x$

Feeling overwhelmed? Just follow these steps every time you need to graph an exponential function.

1. Identify your key values: Find $a$ (the initial value) and $b$ (the base).
2. Determine the behavior: Is it growth ($b>1$) or decay ($0<b<1$)? This tells you the general shape of your curve.
3. Find the Y-Intercept: Plot your starting point. The $y$-intercept is always at $(0,a)$.
4. Create a small table: Choose a few $x$-values around 0 (like $-1,0,1,2$) and calculate the corresponding $y$-values. Remember to follow the order of operations!
5. Plot and Sketch: Plot the points from your table and connect them with a smooth curve. Make sure your curve's shape matches what you determined in Step 2.
6. Draw the Asymptote: Lightly sketch the horizontal asymptote at $y=0$ (the x-axis) as a dashed line to guide your curve. Ensure your graph approaches this line but does not touch it.