Problem 35 - A Graphical Approach to Precalculus with Limits

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 comprehensive graph.

Step 1: Understand the Function

The function given is a linear equation of the form \(f(x) = 4x + 20\). This represents a straight line with a slope of 4 and a y-intercept of 20. A positive slope indicates that the line will rise as it moves from left to right.

Step 2: Setup Window A

Window A is defined by the x- and y-axis ranges \([-10, 10]\). This means your graphing calculator will display the graph within these x-values and y-values on the screen. Any points of the line outside this range will not be visible.

Step 3: Graph the Function in Window A

Entering the function into the graphing calculator and viewing it in Window A, observe the intercepts and slope. The y-intercept of 20 will not be visible since it is outside the y-range of \([-10, 10]\). You will only see a portion of the line.

Step 4: Setup Window B

Window B offers x-axis range \([-10, 10]\) and y-axis \([-5, 25]\). This y-range includes the y-intercept of 20, thereby showing more of the relevant part of the line in context.

Step 5: Graph the Function in Window B

Entering the function into the graphing calculator using Window B, you'll observe the line with the y-intercept at 20 clearly displayed, along with portions of the line both left and right, allowing for a complete view of the function's behavior within the range.

Step 6: Determine the Best Window

Comparing both windows, Window B is more comprehensive as it displays the y-intercept and gives a clear context of how the line behaves within the given domain. It effectively shows how the line crosses the y-axis and moves in both directions.

Key Concepts

Graphing Calculators

Graphing calculators are powerful tools for visualizing linear functions. They help students understand the relationship between equations and their graphs by displaying lines on a coordinate plane.
These devices allow you to input equations, and they automatically generate graphs, providing a visual representation of the function. When you're working with linear equations, like the function in our exercise, this can quickly illustrate how changes in parameters, such as slope and y-intercept, affect the line's appearance.
When using a graphing calculator, it is crucial to choose the right window settings to display the relevant parts of the graph. A window is defined by its x and y-axis ranges.

• Window A: Both x and y values range from ([-10,10]).
• Window B: x ranges from ([-10,10]), whereas y ranges from ([-5,25]).

Choosing suitable window settings ensures that important graph features, like the y-intercept, are displayed clearly. In our exercise, Window B provides a more comprehensive view of the line, including its y-intercept.

Y-intercept

The y-intercept of a linear function is where the graph of the function crosses the y-axis. This is an essential feature because it gives us a specific starting point on the graph.

In mathematical terms, the y-intercept is the point (0,c) in the equation of a line in slope-intercept form (y = mx + c). For the function in our exercise, (4x + 20), the y-intercept is 20. It is the constant term in the equation, showing where the line crosses the y-axis when the x-value is zero. Understanding the y-intercept can help in visualizing and sketching the graph even without a graphing calculator:

• It serves as an anchor point for the graph.
• It illustrates the initial value of the line.

In the context of our original exercise, selecting Window B on the graphing calculator is key as it displays the y-intercept of 20, clearly illustrating this starting point on the graph.

Slope

The slope of a linear function is a measure of its steepness and direction. It describes how much the line rises or falls as it moves horizontally across the graph.

In the general equation of a line (y = mx + c), the slope is represented by (m). For our given function (4x + 20), the slope is 4, indicating the line rises 4 units for every 1 unit it moves to the right. A positive slope, as seen with 4 in our function, means the line moves upward from left to right. A zero slope would mean the line is horizontal, and a negative slope would mean it slopes downward.

• The slope is crucial for predicting the line's direction on a graph.
• It indicates whether a relationship is increasing or decreasing.

Having a clear understanding of slope helps to accurately draw and interpret the trends that linear functions present on a graph.