Problem 112 - Algebra for College Students

Question

Graph \(y=2 x-400\) and \(y=-0.5 x+1\) on the same screen, using the viewing window \(-500 \leq x \leq 500\) and \(-1000 \leq y \leq 1000 .\) Should these lines be perpendicular? Explain.

Step-by-Step Solution

Answer

Yes, the lines are perpendicular because their slopes multiply to -1.

Step 1 - Write Down the Equations

The given equations are: 1. \(y = 2x - 400\) 2. \(y = -0.5x + 1\)

Step 2 - Determine the Slopes

Identify the slopes (m) of each line from the equations: For \(y = 2x - 400\): Slope \(m_1 = 2\) For \(y = -0.5x + 1\): Slope \(m_2 = -0.5\)

Step 3 - Check for Perpendicularity

Two lines are perpendicular if the product of their slopes \(m_1 \times m_2 = -1\). Calculate the product: \(m_1 \times m_2 = 2 \times -0.5 = -1\). Since the result is -1, the lines are perpendicular.

Step 4 - Set Up Viewing Window

Set up the graphing viewing window with \(-500 \leq x \leq 500\) and \(-1000 \leq y \leq 1000\).

Step 5 - Plot the Equations

Graph the lines \(y = 2x - 400\) and \(y = -0.5x + 1\) within the specified viewing window.

Step 6 - Verify the Result

Review the graph to confirm the lines intersect at a right angle, verifying they are perpendicular.

Key Concepts

Slope

When graphing linear equations, understanding the slope is essential. The slope determines the steepness of the line and the direction it moves. In a linear equation of the form \(y = mx + b\), the slope is represented by the coefficient \(m\).

For example, in the equation \(y = 2x - 400\), the slope \(m\) is 2. This means the line rises 2 units vertically for every 1 unit it moves horizontally. This is a positive slope, indicating the line goes upwards.

In contrast, the equation \(y = -0.5x + 1\) has a slope \(m = -0.5\). This means the line goes down by 0.5 units for every 1 unit it moves to the right, representing a negative slope.

Slopes can help predict where the lines will be on the graph and how they will intersect with other lines.

Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90 degrees). Mathematically, this occurs when the product of their slopes is -1. This is expressed as:

• \( m_1 \times m_2 = -1 \)

For the given equations:

• First equation: \( y = 2x - 400 \) with slope \( m_1 = 2 \)
• Second equation: \( y = -0.5x + 1 \) with slope \( m_2 = -0.5 \)

The calculation is straightforward:

• \( 2 \times -0.5 = -1 \)

Thus, these lines are perpendicular. Graphing them will confirm they intersect at a right angle, creating a cross-like intersection.

Viewing Window

Choosing the right viewing window is crucial for accurately plotting equations on a graph. The viewing window sets the range for the x-axis and y-axis, ensuring all important parts of the graph are visible.

For our example, we set the viewing window to

• \( -500 \leq x \leq 500 \)
• \( -1000 \leq y \leq 1000 \)

This wide range allows us to see where the lines intersect, ensuring both the starting points and slopes are fully visible.

Remember, incorrect viewing windows might cut off parts of the graph, making it difficult to analyze the true nature of the lines. Always adjust based on the highest and lowest values your equations can reach within the practical context.