Problem 64 - Algebra and Trigonometry

Question

The manager of a weekend flea market knows from past experience that if she charges \(x\) dollars for a rental space at the flea market, then the number \(y\) of spaces she can rent is given by the equation \(y=200-4 x .\) (a) Sketch a graph of this linear equation. (Remember that the rental charge per space and the number of spaces rented must both be non negative quantities.) (b) What do the slope, the \(y\) -intercept, and the \(x\)-intercept of the graph represent?

Step-by-Step Solution

Answer

The slope is -4, indicating a decrease of 4 spaces per dollar increased. The y-intercept is 200 (max spaces when free), and the x-intercept is 50 (charge for 0 spaces).

Step 1: Define the Linear Equation

The given linear equation is \( y = 200 - 4x \). This equation represents the relationship between the price \( x \) (in dollars) charged for a rental space and the number of spaces \( y \) that can be rented.

Step 2: Determine the Graph Components

To sketch the graph, identify key components: the slope, the y-intercept, and the x-intercept. The slope is -4, the y-intercept (value of \( y \) when \( x = 0 \)) is 200, and the x-intercept (value of \( x \) when \( y = 0 \)) can be found by solving \( 0 = 200 - 4x \).

Step 3: Calculate the X-intercept

Set \( y = 0 \) to find the x-intercept: \( 0 = 200 - 4x \). Solve for \( x \) by adding \( 4x \) to both sides and dividing by 4: \( x = 50 \). Thus, the x-intercept is (50, 0).

Step 4: Sketch the Graph

Draw the graph on a Cartesian plane with \( x \) on the horizontal axis and \( y \) on the vertical axis. Plot the y-intercept (0, 200) and the x-intercept (50, 0). Draw a straight line connecting these points, depicting the linear decrease of \( y \) as \( x \) increases.

Step 5: Interpret the Graph Components

The slope \(-4\) represents the rate at which the number of rental spaces decreases for every additional dollar charged. The y-intercept (200) is the maximum number of spaces rented when the charge is \(0\) dollars. The x-intercept (50) indicates that the charge must be \(50\) dollars to rent 0 spaces; above this price, no spaces are rented.

Key Concepts

Graph Interpretation

Graph interpretation involves understanding what the visual representation of a mathematical equation tells us. In this case, the graph of the linear equation \( y = 200 - 4x \) provides valuable insights. It is a straight line, which shows a consistent relationship between two variables: the price of rental space \( x \) and the number of spaces rented \( y \).

When interpreting graphs, consider the following:

• **Direction and Slope**: The negative slope indicates a downward trend. As the rental price increases, the number of rented spaces decreases.


• **Intercepts**: These points provide specific details about the relationship. By observing where the line crosses the axes, we learn critical thresholds like maximum capacity when pricing is free, and how high you can charge before no one rents a space.

Graphs are useful for predicting outcomes and visualizing complex relationships, turning abstract numbers into concrete understandings.

Slope-Intercept Form

Slope-Intercept form is a way of writing linear equations like \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept.

For the equation \( y = 200 - 4x \), you can see it matches this format:

• **Slope \( m = -4 \)**: This tells us that for each additional dollar charged, the number of rental spaces decreases by 4. This rate of change helps in understanding how sensitive demand is to price changes.


• **Y-intercept \( b = 200 \)**: This is the starting point of the graph, where \( x = 0 \). It conveys that when no rent is charged, all 200 spaces will be filled. This theoretical intercept indicates maximum occupancy.

The clarity of the Slope-Intercept form allows for easy graph plotting and real-world interpretation.

X and Y Intercepts

Intercepts are crucial in understanding how linear equations relate to graph lines. Both the x and y intercepts provide valuable points that define the line's position on the graph.

• **Y-Intercept**: In \( y = 200 - 4x \), the y-intercept is \( (0, 200) \). It shows where the line starts when it's not affected by changes in \( x \) (pricing). Essentially, it signals the potential maximum use of space if rental fees are waived.


• **X-Intercept**: Found by setting \( y = 0 \), solving gives \( x = 50 \). Mathematically represented as \( (50, 0) \), this indicates that charging \( 50 \) dollars results in zero spaces rented. This intercept tells us beyond this price point, occupancy becomes zero, and is crucial for understanding pricing limits.

These intercepts not only symbolically map out the equation but inform practical decisions like pricing strategies and understanding limitations.