Problem 35 - A Graphical Approach to College Algebra

Question

Graph each linear function on a graphing calculator, using the two different windows given. State which window gives a comprehensive graph. $$f(x)=4 x+20$$ Window A: \([-10,10]\) by \([-10,10]\) Window B: \([-10,10]\) by \([-5,25]\)

Step-by-Step Solution

Answer

Window B gives a more comprehensive graph.

Step 1: Understanding the Function

The function given is a linear function, written as \( f(x) = 4x + 20 \). This defines a straight line with a slope of 4 and a y-intercept of 20. The slope indicates the line rises 4 units for every 1 unit it moves to the right.

Step 2: Setting Up Window A

In Window A, the x-axis ranges from -10 to 10 and the y-axis ranges from -10 to 10. This means the graph will only show y-values from -10 to 10.

Step 3: Evaluating Window A

Calculate the y-values for the x-limits to understand the graph's fit. When \( x = -10 \), \( f(x) = 4(-10) + 20 = -20 \) and when \( x = 10 \), \( f(x) = 4(10) + 20 = 60 \). Since both y-values (-20 and 60) fall outside the y-range \([-10, 10]\), Window A will not display the function properly.

Step 4: Setting Up Window B

In Window B, the x-axis ranges from -10 to 10, while the y-axis ranges from -5 to 25. This window has a greater range for y-values compared to Window A.

Step 5: Evaluating Window B

Using the same x-values as before, we see \( f(-10) = -20 \) and \( f(10) = 60 \). Although the lower y-value is still off the chart, the widened upper y-bound of 25 ensures more of the graph is visible, capturing y-values from 20 upwards, providing a better view of the function's behavior.

Step 6: Determine the Comprehensive Window

Given that more of the graph's range fits within the y-values presented in Window B, while Window A does not display either endpoint adequately, it is clear Window B offers a more comprehensive view of the function.

Key Concepts

Graphing Calculator

A graphing calculator is a crucial tool in understanding and visualizing mathematical functions. It allows you to plot graphs and solve equations with ease. This tool is highly effective for graphing linear functions like the one given in this exercise because it lets you input functions directly and observe their behaviors graphically. Some benefits of using a graphing calculator include:

• Visualizing the slope and intercept of your function, which are key components of its behavior.
• Easily adjusting window settings to find a suitable view of the graph. This is critical as different window settings can significantly alter how a graph appears.
• Checking the graph against algebraic solutions to ensure accuracy.

When working with linear functions, graphing calculators provide students with visual insight that aids comprehension and enhances learning.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\). Here, \(m\) represents the slope and \(b\) is the y-intercept. For our function, \(f(x) = 4x + 20\), the slope is 4, and the y-intercept is 20.Understanding these elements is vital:

• Slope (\(m\)): It tells us how steep the line is. A slope of 4 means for every increase in x by 1 unit, y increases by 4 units.
• Y-intercept (\(b\)): This is where the line crosses the y-axis. In this case, it's 20, where the function starts when \(x = 0\).

Knowing the slope and y-intercept helps in predicting and drawing the graph accurately before seeing it on the display.

Graph Window Settings

Adjusting graph window settings on a calculator influences how a function is displayed. These settings determine the range of x and y values shown on the graph, which in turn affects visibility.For function \(f(x) = 4x + 20\):

• Window A: The y-axis ranges from -10 to 10, but the y-values calculated as -20 and 60 are outside this range. Thus, the function will not be displayed correctly.
• Window B: The y-axis ranges from -5 to 25, allowing a better fit for the y-values. Though -20 is still excluded, the graph displays more of the line, offering a clearer picture.

Choosing the right window settings ensures that the graph aligns with the expected behavior of the function. Always check both axes settings to ensure comprehensive graph coverage.

Function Graphing

Graphing functions allows us to visually interpret mathematical relationships and equations. For the linear function \(f(x) = 4x + 20\), graphing can provide a clear representation of its characteristics.Steps for accurate function graphing include:

• Identifying the slope and intercept using the slope-intercept form. This helps quickly sketch the graph mentally.
• Setting an adequate graph window that covers essential parts of the function.
• Reviewing and interpreting the graph displayed to ensure it behaves as expected based on calculated values.

By following these steps and using graphing calculators, one can easily visualize linear functions and understand how different components like slope and intercept influence the graph's shape and direction.