3.4: Graphs of Exponential Functions
Suppose you were listening to the radio, and heard the DJ announce a call-in giveaway.
"The 343rd caller RIGHT NOW will win $15,000!"
You grab your cell phone and call in right away: "Hi, this is me... did I win?"
"Congratulations! You are today's big winner on WGAM radio!"
"Umm... Really?"
"Just one question for you: 'We Give Away Money' radio listeners want to know, what are you going to do with the money?"
- What would you do? Spend it? Invest it?
If you could get a 5% interest rate, and saved it for your 50th birthday, would it really be all that much, even if you left it there for 30 years or so?
Watch This
Embedded Video:
- James Sousa: Graph Exponential Functions
Guidance
Exponential function graphs are often more useful than specific solutions of exponential functions. Because exponential functions are commonly used to model things like population growth, stock behavior, and temperature changes, it may well only be necessary to get a good general idea of the behavior of the function. In fact, in such situations you will find that solving the function for specific values is often only done in order to graph them, and it is the overall behavior that is important.
Let’s consider the graph of f(x) = 2^{x}. The graph below shows this function, with several points marked in blue.
Notice that as x approaches \begin{align*}\infty\end{align*}, the function grows without bound. That is, \begin{align*}\lim_{x \to \infty} (2^x) = \infty\end{align*}. However, if x approaches \begin{align*}-\infty\end{align*}, the function values get closer and closer to 0. That is, \begin{align*}\lim_{x \to -\infty} (2^x) = 0\end{align*}. Therefore the function is asymptotic to the x-axis. This is the graphical result of the fact that the range of the function is limited to positive y values.
Graphing Exponential Functions Using Transformations
From your prior studies of function transformations, you should recognize the graph of g(x) = 2^{x} + 3 as a vertical shift of the graph of f(x) = 2^{x}. In general, we can produce a graph of an exponential function with base 2 if we analyze the equation of the function in terms of transformations. The table below summarizes the different kinds of transformations of f(x) = 2^{x}.
Equation | Relationship to f(x)=2^{x} | Range |
---|---|---|
\begin{align*}g(x)=\frac{2^x}{2^a}=2^{x-a}, \text{for } a>0\end{align*} | Obtain a graph of g by shifting the graph of f a units to the right. | y > 0 |
\begin{align*}g(x)=2^{a} \cdot 2^{x}=2^{a+x}, \text{for } a>0\end{align*} | Obtain a graph of g by shifting the graph of f a units to the left. | y > 0 |
\begin{align*}\,\! g(x)=2^x+a, \text{for } a>0\end{align*} | Obtain a graph of g by shifting the graph of f up a units. | y > a |
\begin{align*}\,\! g(x)=2^x-a, \text{for } a>0\end{align*} | Obtain a graph of g by shifting the graph of f down a units. | y > a |
\begin{align*}\,\! g(x)=a(2^x), \text{for } a>0\end{align*} | Obtain a graph of g by vertically stretching the graph of f by a factor of a. | y > 0 |
\begin{align*}\,\! g(x)=2^{ax}, \text{for } a>0\end{align*} | Obtain a graph of g by horizontally compressing the graph of f by a factor of a. | y > 0 |
\begin{align*}\,\! g(x)=-2^x\end{align*} | Obtain a graph of g by reflecting the graph of f over the x-axis. | y > 0 |
\begin{align*}\,\! g(x)=2^{-x}\end{align*} | Obtain a graph of g by reflecting the graph of f over the y-axis. | y > 0 |
Example A
Use a graphing utility to graph f(x) = 2^{x}, g(x) = 3^{x} and h(x) = 4^{x}. How are the graphs the same, and how are they different?
Solution:
f(x) = 2^{x}, g(x) = 3^{x} and h(x) = 4^{x} are shown together below.
The graphs of the three functions have the same overall shape: they have the same end behavior, and they all contain the point (0, 1). The difference lies in their rate of growth. Notice that for positive x values, h(x) = 4^{x} grows the fastest and f(x) = 2^{x} grows the slowest. The function values for h(x) = 4^{x} are highest and the function values for f(x) = 2^{x} are the lowest for any given value of x. For negative x values, the relationship changes: f(x) = 2^{x} has the highest values of the three functions.
Example B
Use transformations to graph the function a(x) = 3^{x} ^{+ 2}
Solution:
a(x) = 3^{x} ^{+ 2}
This graph represents a shift of y = 3^{x} two units to the left. The graph below shows this relationship between the graphs of these two functions:
Example C
Use transformations to graph the function b(x) = -3^{x} + 4
Solution:
b(x) = -3^{x} + 4
This graph represents a reflection over the y-axis and a vertical shift of 4 units. You can produce a graph of b(x) using three steps: sketch y = 3^{x}, reflect the graph over the x-axis, and then shift the graph up 4 units. The graph below shows this process:
While you can always quickly create a graph using a graphing utility, using transformations will allow you to sketch a graph relatively quickly on your own. If we start with a parent function such as y = 3^{x}, you can quickly plot several points: (0, 1), (2, 9), (-1, 1/3), etc. Then you can transform the graph, as we did in the previous example.
Notice that when we sketch a graph, we choose x values, and then use the equation to find y values.
Remember the problem at the beginning of the lesson? Were you able to solve it? Is a 5% return on a $15,000 investment really worth all that much over 30 years? You bet it is, look at the graph below:
Note that the y axis denotes 'thousands'. After 30 years, the original $15k investment becomes nearly $65,000, not a bad birthday present for yourself! |
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Vocabulary
An exponential function is a function where the input variable (the x term) is found in the exponent, e.g.: \begin{align*}y = 3^{x}\end{align*}.
If a function grows without bound, it has no limit (it stretches to \begin{align*}\infty\end{align*}).
An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).
Guided Practice
Questions 1) Sketch the graph of the equation
- \begin{align*}y = 2(\frac{1}{2}x + 3)^2 - 1\end{align*}
2) Write an equation for the graph below:
3) Describe the transformation from the parent function:
- \begin{align*}y = 6(9^x) + 9\end{align*}
4) Describe the Transformation and graph:
- \begin{align*}y = -5|-4(x + 3)| - 2\end{align*}
Answers
1) Break the parabola equation down piece-by-piece:
- In front of the parentheses we have a 2, so this function is stretched by a factor of 2.
- The co-efficient of the x is horizontal compression, in this case, we'll be stretching by a factor of 2
- The -6 inside the parenthesis is horizontal shift, so this graph moved left 6.
- The number outside the parenthesis is vertical shift, so this graph moved down 1
2) \begin{align*}y = -3|x -2| +3\end{align*}
- The negative 3 outside of the absolute value means the graph is flipped upside down, and then stretched by a factor of 3.
- The number inside the absolute value is horizontal shift, so this graph moved right 2
- The number outside the absolute value is vertical shift, so this graph moved up 3
3) The absolute value of the number in front of the function is vertical stretch: our graph has a vertical stretch of 6.
- The number outside the parenthesis is vertical shift, so we move the graph up 9.
4) The number in front of the function is negative, so this graph is reflected over the x-axis
- The value of the number in front of the function is vertical stretch, so this graph has a vertical stretch of 5.
- The coefficient of the x is negative: the graph is reflected over the y-axis.
- The value of the x coefficient, is the horizontal compression. This graph has a horizontal compression of 4.
- The number inside the parenthesis is horizontal shift (positive value, so shift left)
- The number outside the parenthesis is vertical shift, this graph has a shift DOWN.
- Therefore our answer is: Reflected over the x-axis. Vertical stretch of 5. Reflected over the y-axis. Horizontal stretch of 1/4 . Left 3. Down 2.
The Graph looks like this:
Practice
Sketch the graph of each function.
- \begin{align*}y = 2 \cdot 3^x\end{align*}
- \begin{align*}y = 4 \cdot \frac{1}{2}^2\end{align*}
- \begin{align*}y = 2 \cdot \frac{1}{2}^{x+1} + 2\end{align*}
Write an equation for each graph.
Use graphing transformation rules to make a conjecture about what the graph of each function will look like.
- \begin{align*}f(x) = 3^{x-4} \end{align*}
- \begin{align*}f(x) = -4^x\end{align*}
- \begin{align*}f(x) = 3^x - 2\end{align*}
- \begin{align*}f(x) = -5^{x +2}\end{align*}
- \begin{align*}f(x) = 5^{x-4} - 3\end{align*}
Graph the following functions.
Describe the transformations applied to the parent graph \begin{align*}f(x) = n^x\end{align*} to obtain the graph of each function.
- \begin{align*} g(x) = \frac{1}{3} (2^x)\end{align*} Transformation:
- \begin{align*} m(x) = -2^{3x}\end{align*} Transformation:
- \begin{align*} s(x) = 2^{x-5}\end{align*} Transformation:
- \begin{align*}t(x) = -3(2)^{x-5} - 4\end{align*} Transformation:
Asymptotes
An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).Asymptotic
A function is asymptotic to a given line if the given line is an asymptote of the function.Exponential Function
An exponential function is a function whose variable is in the exponent. The general form is .grows without bound
If a function grows without bound, it has no limit (it stretches to ).Horizontal Asymptote
A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.Transformations
Transformations are used to change the graph of a parent function into the graph of a more complex function.Image Attributions
Here you will explore the graphs of exponential functions using transformations and technology.