Problem 98 - College Algebra

Question

For the following exercises, use a calculator or graphing technology to complete the task. Graph the function \(f\) on a domain of [-10,10]\(: f(x)=0.02 x-0.01\). Enter the function in a graphing utility. For the viewing window, set the minimum value of \(x\) to be -10 and the maximum value of \(x\) to be 10 .

Step-by-Step Solution

Answer

Graph the function by entering \( f(x) = 0.02x - 0.01 \) into a graphing tool with \(x\) domain set to [-10, 10].

Step 1: Understand the Function

We need to graph the linear function \( f(x) = 0.02x - 0.01 \) over the domain \([-10, 10]\). This function represents a straight line with a slope of \(0.02\) and a y-intercept of \(-0.01\).

Step 2: Set Up the Graphing Utility

Open a graphing calculator or software and locate the function input area. Ensure the viewing window covers the entire specified domain of \(x = -10\) to \(x = 10\).

Step 3: Enter the Function

Input the function \( f(x) = 0.02x - 0.01 \) into the graphing utility. Make sure it is entered correctly without any syntax errors.

Step 4: Adjust the Viewing Window

Set the viewing window to have the \(x\)-axis ranging from \(x = -10\) to \(x = 10\). Optionally, adjust the \(y\)-axis to have a reasonable range to clearly see the graph; for example, \(-1\) to \(1\).

Step 5: Graph the Function

Press the 'Graph' or 'Plot' button in your utility to visualize the function. Check to ensure the line extends between the points \(x = -10\) and \(x = 10\) exactly as specified.

Key Concepts

Using a Graphing Utility Effectively

Graphing utilities are powerful tools for visualizing mathematical functions. Whether you're using a physical calculator or software on your computer, they help bring equations to life. Launching a graphing utility often involves locating the function entry bar where you'll input your equation. This is typically straightforward and allows for entering linear equations like any typed numerical expression.

• Ensure window settings match your intended range, such as scaling from -10 to 10 on the x-axis, as specified in your exercise.
• An appropriate view ensures the behavior of the graph is clearly visible and easily interpreted.
• Utilize tools within the utility to zoom or adjust to see important aspects of the graph better.
• Software often allows you to input several equations at once, comparing graphs side by side.

Using these tools effectively not only enhances understanding but also lets you test and explore further scenarios confidently.

Understanding Linear Equations

Linear equations form the basis of many mathematical models and are defined by their constant rate of change. The general form of a linear equation is expressed as \( y = mx + b \), where \( m \) indicates the slope, and \( b \) represents the y-intercept. In the equation \( f(x) = 0.02x - 0.01 \), these components are:

• Slope \( m = 0.02 \): Indicates the line rises very slightly with every increase in \( x \), emphasizing a gradual incline in this scenario.
• Y-intercept \( b = -0.01 \): Describes where the line crosses the y-axis, just below the origin in this case.

Understanding these terms helps in deciphering what the line represents in real-world contexts. Linear equations are critical because they describe relationships that can be easily interpreted and applied to numerous scenarios. These equations are often used to show direct proportional relationships, making them essential in fields like physics, economics, and everyday problem-solving.

Concept of Domain and Range in Functions

The domain and range are fundamental concepts describing the input and output of a function, respectively. When graphing, these define what parts of your graph are drawn and where they lie within the Cartesian plane. For the equation \( f(x) = 0.02x - 0.01 \) with a domain of \([-10, 10]\), this means:

• Domain \([-10, 10]\): This is the set of all possible input values of \( x \), creating boundaries on the horizontal axis.
• Range: It comprises all possible output values (or \( y \)-values) the equation can produce from the domain.

For a given domain, calculate the smallest and largest \( y \)-values to understand the range. By substituting \( x = -10 \) and \( x = 10 \) back into the equation, you find corresponding \( y \)-values to help visualize the extent of the graph within these bounds. Understanding domain and range allows a clearer understanding of how a function behaves throughout its entire scope.