Exponential+Functions-Target+A-Exponentially+Speaking-Guided+Learning

Target A: Graph exponential functions and identify the domain and range.

 * [[image:Discuss this.png width="173" height="97"]] || Option 1: You are paid $1000 a year for 20 years.

Option 2: You are paid 1 dollar the first year, 2 dollars the second year , 4 dollars for the third year, doubling the amount each year.

Which option would you choose and why? ||

Let's graph an exponential function and see what the graph will look like. We have not yet graphed a function of this form. We will be looking to see what the graph looks like. Will it look like a linear function? What will the domain be? What will the range look like? What will the end behavior of the graph be? math f(x)=a\cdot b^{x} math  = //a// ⋅ //b//    //x//                //f// ( //x// <span class="mo" style="font-family: MathJax_Main;">) <span class="mo" style="font-family: MathJax_Main; padding-left: 0.26em;">= <span class="mi" style="font-family: MathJax_Math; padding-left: 0.26em;">//a// <span class="mo" style="font-family: MathJax_Main; padding-left: 0.196em;">⋅  <span style="clip: rect(1.606em 1000.003em 2.696em -0.51em); left: 0.003em; position: absolute; top: -2.497em;"><span class="mi" style="font-family: MathJax_Math;">//b//    <span class="mi" style="font-family: MathJax_Math; font-size: 70.7%;">//x// An exponential function has the following restrictions: math a\neq 0, b>0,b\neq 1 math What do these restrictions mean to us?

math a\neq 0 math The co nstant // a // cannot be zero because if // a = //0 we are looking at the equation // f // ( // x // ) = 0, because 0 multiplied by anything would be 0. This would make the graph into a horizontal line (remember // f // ( // x // ) is the same as // y //, so // y // = 0). We are supposed to be graphing an exponential function, not a horizontal line!

math b>0 math The base //b// has to be greater than zero because if it is not greater than zero we could potentially end up evaluating the math \sqrt{\text{-}2} math because of exponents that are rational and we know if we try to evaluate this it is not a real number!

math b\neq 1 math The base //b// cannot equal 1 because again this would put us in a situation where we would be graphing a horizontal line of //f//(//x//) = //a//. If //b// = 1 we would just be mutilplying a by 1 so we would end up with the equation //f//(//x//) = (1)(//a//) = //a//, where //a// is just a constant or a number - so the graph would be a horizontal line.

math f(x)=2^{x} math Notice //a// = 1 and //b// = 2. We will choose to use a table to graph the function so we can see what shape this graph will take. When we are choosing our domain (//x//) there is no restriction on the domain that are mentioned so we will be able to choose any real number. Let's choose to use some negative numbers, positive numbers and zero for our domain. Watch the following video to see how to graph this basic exponential function. media type="custom" key="27721163"
 * Example 1: Graph the exponential function**

As you can see, this is not linear it has a curve to it - it is exponential. As was mentioned in the video, we see that it has what is called an asymptote. This is a line that the graph of the function will never cross. The function will come very close but never will cross this line. The asymptote of this particular function is the //x//-axis or //y// = 0. As we also noticed, this asymptote defined the range of the function. The range will be //y// > 0. All the y values of the function will be larger than zero. Also in the video, end behavior was discussed. We look at the ends of the graph to see what is happening to the //y// values as //x// approaches positive infinity, very large numbers, as well as what happens to the y values as //x// approaches negative infinity, very small numbers. This is considered a growth model because as the x-values increase, the //y// - values are increasing. Also, let's make sure to notice that the //y//-intercept is (0, 1).

math f(x)=(\frac{1}{2})^{x} math
 * Example 2: Graph the exponential function**

Let's graph this exponential where the base, //b,// is a fraction to see what it's graph looks like.

media type="custom" key="27721167"

Notice with this exponential function, the domain remained the same. It still has an asymptote at the //x//-axis or //y// = 0. The graph looks different from the first example and the end behavior is different. This particular exponential function is a decay model. The //y//-values are decreasing as the //x//-values are increasing. Another point we should look at is the //y//-intercept. Notice it is (0,1), similar to the graph in **Example 1**.

math f(x)=\frac{1}{2}\cdot 4^{x} math
 * Example 3: Graph the exponential function: **

This time let's see what happens when the coefficient, //a,// is not equal to 1 - we will see it's affect on the graph.

media type="custom" key="27721171"

What did that //a// value actually do? Notice in the last two examples our //y//-intercept was at (0, 1) when our //a// value was equal to one. Notice this time the //y//-intercept is (0, 1/2) and our //a// value was equal to 1/2! Aha - you should notice that the //y//-intercept will be the point (//a//, 0) on the graph of an exponential function.

This section is just the beginning of our discussion of exponential function graphs. It should give you a good taste of the behavior of these graphs.

How does the average rate of change of an exponential function compare to the average rate of change of a linear function? Use the sliders "a" and "b" and the points A and B to answer the questions below. math f(x)=a \cdot b^x math media type="custom" key="27721181"
 * Average Rate of Change:**

Graph the following exponential functions. Make sure to state the domain, range and end behavior.
 * Quick Check - Exponential Functions - Target A - Exponentially Speaking **

math a) f(x)=3^{x} math

math b) h(x)=-2^{x}

math

Quick Check Solution - Exponential Functions - Target A - Exponentially Speaking