Problem 98 - College Algebra

Question

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 on domain \([-10, 10]\); it's a line with slope 0.02.

Step 1: Understand the Function

We are given the function \( f(x) = 0.02x - 0.01 \). This is a linear function, which means its graph will be a straight line. The function is defined on the domain \([-10, 10]\).

Step 2: Determine the Slope and Y-intercept

For the function \( f(x) = 0.02x - 0.01 \), the slope is \(0.02\) and the y-intercept is \(-0.01\). This tells us how steep the line is and where it crosses the y-axis.

Step 3: Set Up the Graphing Utility

Input the function \( f(x) = 0.02x - 0.01 \) into a graphing utility. Set the x-axis to display the range from \(-10\) to \(10\). You may set the y-axis range as the utility's default or adjust according to the scale needed.

Step 4: Graph the Function

Once entered and the viewing window is adjusted, plot the function. You should see a straight line crossing the y-axis at \(-0.01\) and having small positive slope, meaning it slightly inclines to the right as x increases.

Key Concepts

Domain and Range

When exploring linear functions, understanding the domain and range is crucial. The domain of a function is the set of all possible input values—essentially, the 'x' values that you can put into the function. In this exercise, the domain is defined as \([-10, 10]\), which means you can choose any x-value between -10 and 10, inclusive, to plug into the function. These boundaries are important, especially when using a graphing utility because it tells the tool where to start and stop displaying the line.The range, on the other hand, refers to all possible output values—essentially, the 'y' values produced by plugging domain values into the function. In linear functions like this one, the range is directly influenced by the domain. As the x-value changes across the domain, the corresponding y-value will vary. In this case, the function will output values starting at the lower boundary of the domain up to its maximum. Due to the linearity of the function, you see a direct and continuous relationship between x and y-values.

Linear Equations

Linear equations are expressions that form straight lines when graphed. They are typically arranged in the format \( f(x) = mx + b \), where \(m\) represents the slope and \(b\) represents the y-intercept. In our exercise, the linear equation is \( f(x) = 0.02x - 0.01 \). This simple format helps in predicting how the graph will look.In essence, linear equations demonstrate a constant rate of change. This means that for each unit increase in x, there will be a consistent change in y. That's why, when plotted, these equations always result in straight lines. They are essential tools in mathematics and various fields because they help illustrate relationships between variables clearly.

Slope and Y-intercept

Understanding the slope and y-intercept in the context of a linear function is fundamental. The slope, denoted as \( m \) in the equation \( f(x) = mx + b \), signifies the rate at which y changes with respect to x. For our function \( f(x) = 0.02x - 0.01 \), the slope is \( 0.02 \). This indicates that for every unit increase in x, y increases by \0.02\. A positive slope, like this one, suggests that the line slopes up to the right, although here it's quite gradual.The y-intercept, marked as \( b \), indicates the point at which the line crosses the y-axis. In this case, the y-intercept is \( -0.01 \). This means when \( x = 0 \), the function value \( f(x) \) is \( -0.01 \). Graphically, this is where the line meets the vertical axis. Always remember that the slope shows the direction and steepness, while the y-intercept pinpoints the line's starting position on the y-axis. These two together define the unique position and angle of the line on a graph.