Problem 2 - Thinking Mathematically

Question

Use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). \(f(x)=5^{x}\)

Step-by-Step Solution

Answer

The coordinates for the exponential function \(f(x) = 5^x\) given the x-values -2,-1,0,1,2 are (-2,0.04), (-1,0.2), (0,1), (1,5), and (2,25)

Step 1: Understand the exponential function

In this exercise, the exponential function is \(f(x) = 5^x\). Here, the base is 5 and the exponent is x. For any value of x that you input, the output will be 5 raised to the power of that value.

Step 2: Construct the table of coordinates

Create a table with two columns, one for x and another for \(f(x)\). Then fill in the given x-values which are -2,-1,0,1, and 2.

Step 3: Compute the corresponding output for each input

Now calculate \(f(x)\) using the given values of x in the function \(f(x) = 5^x\).\nFor \(x=-2\), \(f(-2)=5^{-2}=0.04\)\nFor \(x=-1\), \(f(-1)=5^{-1}=0.2\)\nFor \(x=0\), \(f(0)=5^{0}=1\)\nFor \(x=1\), \(f(1)=5^{1}=5\)\nFor \(x=2\), \(f(2)=5^{2}=25\)

Step 4: Plot the coordinates

The last step is to plot these coordinates onto a graph to visualize the exponential function. Every pair of (x, \(f(x)\)) values represents a point on the graph.

Key Concepts

Understanding Graphing Techniques for Exponential Functions

Graphing an exponential function can really help in visualizing how it behaves across various values. When you graph an exponential function, like \(f(x)=5^x\), you are essentially showing the relationship between the input \(x\) and the output \(f(x)\). To start graphing, a common technique is to select specific x-values to get a clearer picture of the graph's shape. In our exercise, we use the x-values -2, -1, 0, 1, and 2.

Once you have the values of \(x\), you calculate the corresponding \(f(x)\) using the function formula. For instance, with \(f(x) = 5^x\), you compute results such as \(5^{-2} = 0.04\) and \(5^2 = 25\). It's like connecting the dots—once you've plotted these pairs on the graph, you can draw a smooth curve through them. This curve gives you a visual representation of the exponential function.

Graphing such exponential equations allows you to see patterns at a glance, like how the function rapidly increases as \(x\) grows. This visual technique helps immensely in understanding and analyzing mathematical functions.

Creating and Interpreting Coordinate Tables

A coordinate table is a critical tool when graphing mathematical functions. It organizes values so that you can systematically approach creating a graph. For an exponential function like \(f(x) = 5^x\), the table includes two columns: one for the chosen \(x\) values and another for the corresponding function outputs, \(f(x)\).

This systematic approach helps prevent mistakes, as you can carefully calculate each \(f(x)\) result for every \(x\) value. You can start with simple values like -2, -1, 0, 1, and 2, which often reveal essential trends in the function. For example, when \(x = 1\), \(f(x) = 5\), illustrating how quickly the function can increase.

These tables not only organize information but also enhance understanding, as seeing the growth in numbers helps grasp the function's exponential nature. Such tables are indispensable for students or anyone trying to learn more about mathematical functions.

Defining and Analyzing Mathematical Functions

Mathematical functions like \(f(x) = 5^x\) are expressions that relate an input \(x\) to a single output. Exponential functions have a specific form where the input, or variable, is an exponent. This form leads to rapid changes in function values, illustrating growth or decay.

An exponential function with base greater than 1, such as \(5^x\), results in a distinctly increasing curve on a graph. This helps explain why exponential functions are found in many natural phenomena, like population growth and radioactive decay, where things change quickly and continuously.

Understanding mathematical functions involves learning their properties and effects. For example, choosing the correct base and recognizing the effect of different values can alter the function's behavior and appearance on a graph. Being familiar with these principles enables learners to predict patterns and solve complex problems more effectively.