Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the functions \(f, g,\) and \(h\) in the same viewing window. $$f(x)=\frac{1}{3} x, \quad g(x)=-x+4, \quad h(x)=f(x)-g(x)$$

Step-by-Step Solution

Answer

\nThe graph of \(f(x)\) is a line with positive slope 1/3 passing through the origin, \(g(x)\)'s graph is a line with negative slope -1 and y-intercept 4, and \(h(x)\) is the result of subtracting \(g(x)\) from \(f(x)\), which should also be plotted on the graph.

Step 1: Graph function f(x)

First, we graph the function \(f(x) = \frac{1}{3}x \). This is a straight line equation, where 1/3 is the slope of the line. For this function, the line will rise one unit vertically for every three units horizontally and pass through the origin.

Step 2: Graph function g(x)

Next, we graph the function \(g(x)= -x + 4\). This function is also a straight line. The -1 in front of the x indicates that the slope is negative, meaning the line will descend as it moves to the right. The number 4 is the y-intercept or where the line crosses the y-axis.

Step 3: Calculate and graph function h(x)

Finally, we calculate the function \(h(x) = f(x) - g(x)\). This requires us to subtract the equation for g(x) from the equation for f(x). Once calculated, h(x) should be graphed on the same plane.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra that describe a straight line when plotted on a graph. Each equation can be written in the form \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.

A linear equation always produces a straight line graph.

The term linear comes from the Latin word "linea," meaning a line.

Understanding linear equations is essential because they reveal the relationship between two variables in a predictable manner. This predictability makes linear equations useful in many real-world applications, such as calculating distance, budgeting, and even forecasting trends.
Mathematically, when we say a function is linear, we mean that for any two points on the line, the change in the y-values divided by the change in the x-values is constant. This constant is the slope of the line.

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a line. It has the format \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept:

• **Slope (m):** Determines the steepness of the line. If \(m\) is positive, the line ascends from left to right. If \(m\) is negative, it descends.
• **Y-intercept (b):** The point where the line crosses the y-axis. It represents the value of \(y\) when \(x\) is zero.

The slope-intercept form is beneficial because it provides both the direction and starting point of the line at a glance. For example, in the equation \(g(x) = -x + 4\), the slope of \(-1\) indicates the line is declining as it moves to the right, while a y-intercept of 4 shows it crosses the y-axis at y = 4.
To convert any linear equation into this form, ensure it is solved for \(y\). This representation then makes graphing straightforward and intuitive.

Graphing Utility

A graphing utility is a versatile tool that allows you to visualize mathematical functions by plotting them on a coordinate plane. These utilities can be physical calculators with screens, computer software, or online applications that produce detailed graphs.
Using a graphing utility simplifies the process of seeing how different functions compare against one another. For instance, observing \(f(x)\), \(g(x)\), and \(h(x)\) together can help identify intersections, parallel lines, and how each line behaves across the plane.
Graphing utilities often allow you to:

• Input different equations to see their graph representations.
• Adjust the viewing window to focus on points of interest.
• Zoom in or out to gain valuable insights into function behavior.

By using these tools, understanding complex relationships between linear equations becomes more approachable. They turn abstract numbers into a tangible image, helping students and professionals alike to decipher and analyze data efficiently.