Problem 44 - Intermediate Algebra

Question

Suppose that the daily profit from an ice cream stand is given by the equation \(p=2 n-4\), where \(n\) represents the gallons of ice cream mix used in a day and \(p\) represents the dollars of profit. Label the horizontal axis \(n\) and the vertical axis \(p\), and graph the equation \(p=2 n-4\) for nonnegative values of \(n\).

Step-by-Step Solution

Answer

Plot points (0, -4) and (2, 0) for graphing. The profit becomes positive for \( n \geq 2 \).

Step 1: Understand the Equation

We are given that the profit, \( p \), is calculated by the equation \( p = 2n - 4 \). Here, \( n \) represents the number of gallons of ice cream mix used, and \( p \) is the daily profit in dollars. This is a linear equation of the form \( p = mn + c \) where \( m = 2 \) is the slope and \( c = -4 \) is the y-intercept.

Step 2: Find Key Points

To graph the line, find at least two key points. Start by finding the y-intercept by setting \( n = 0 \):- When \( n = 0 \), \( p = 2(0) - 4 = -4 \). So, the point is (0, -4).Next, find the x-intercept by setting \( p = 0 \):- Solve \( 0 = 2n - 4 \) to get \( 2n = 4 \) so \( n = 2 \). The point is (2, 0).

Step 3: Plot the Intercepts

On a coordinate plane with \( n \) on the x-axis and \( p \) on the y-axis, plot the points (0, -4) and (2, 0). These are the intercepts of the line and will help determine the linear relation between \( n \) and \( p \).

Step 4: Draw the Graph

Using a ruler, draw a straight line through the points (0, -4) and (2, 0). Since \( n \) represents gallons used and cannot be negative, ensure the line is only drawn for \( n \geq 0 \). This part of the line should extend right indefinitely as \( n \) increases.

Step 5: Label the Axes and Interpret

Label the horizontal axis as \( n \) (gallons of ice cream) and the vertical axis as \( p \) (dollars of profit). The graph shows that as \( n \) increases, the profit increases linearly. Profit is negative when less than 2 gallons are used but becomes positive when \( n \geq 2 \).

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that helps us visualize relationships between variables. In this exercise, the equation \( p = 2n - 4 \) represents a line on a graph where \( n \) is the number of gallons of ice cream mix, and \( p \) is the profit in dollars. To graph this equation:

• Understand it as a straight line with the form \( p = mn + c \), where \( m \) acts as the slope and \( c \) as the y-intercept.
• Identify two crucial points the line passes through, such as the intercepts.
• Plot these points on a coordinate system, with the horizontal axis as \( n \) and the vertical axis as \( p \).

Once plotted, draw a line through these points to show the linear relationship. This provides a visual representation of the equation, indicating how profit changes with ice cream mix usage.

Intercepts

Intercepts are specific points where the line crosses the axes. They provide essential information about the line without solving the whole equation. There are two types of intercepts:

• Y-Intercept: This is where the line crosses the vertical axis (\(p\)-axis in this case). You find it by setting \(n = 0\). For our equation, when \(n = 0\), \(p = 2 \times 0 - 4 = -4\). So, the y-intercept is the point (0, -4).
• X-Intercept: This is where the line crosses the horizontal axis (\(n\)-axis). Set \(p = 0\) to find it. Solving \(0 = 2n - 4\) gives \(n = 2\), thus the x-intercept is (2, 0).

These intercepts are simple to calculate and provide pivotal reference points for graphing the line.

Slope

In the context of linear equations, the slope is a measure of steepness and direction. It defines how much \( p \), the output, changes as \( n \), the input, increases by one unit. The slope is noted as \( m \) in the line equation \( p = mn + c \). Here, \( m = 2 \), meaning for each additional gallon of mix, the profit increases by $2.
Understanding slope helps you:

• Determine whether a line is increasing or decreasing. A positive slope like \( 2 \) means the line rises as it moves to the right.
• Compare different linear relationships by examining their steepness.

When sketching the graph, ensure the line reflects this constant rate of change across all points. This effectively demonstrates the relationship between \( n \) and \( p \).