Question

Plot each pair of points and determine the slope of the line containing the points. Graph the line. $$ (2,3) ;(4,0) $$

Short answer

The slope is \( -\frac{3}{2} \).

Step-by-step solution

Plot the Points

Plot the points (2,3) and (4,0) on a Cartesian coordinate plane. Locate (2,3) by moving 2 units to the right (positive x-direction) and 3 units up (positive y-direction). Then locate (4,0) by moving 4 units to the right (positive x-direction) and 0 units up (staying on the x-axis).

Calculate the Slope

Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) where \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (4, 0) \). Substitute the given points into the formula: \[ m = \frac{0 - 3}{4 - 2} = \frac{-3}{2} = -\frac{3}{2} \]

Graph the Line

Draw a straight line through the points (2,3) and (4,0) on the coordinate plane. Since the slope is \( -\frac{3}{2} \), it means the line falls 3 units for every 2 units it moves to the right. The graph should show this descending line.

Key concepts

Plotting Points

Before we can determine the slope and graph the line, we need to plot the points on the Cartesian coordinate plane. Let's start with our given points: (2,3) and (4,0). To plot (2,3), move 2 units to the right along the x-axis and then 3 units up on the y-axis. You should now be at the coordinates (2,3). Next, to plot (4,0) move 4 units to the right and remain on the x-axis since the y-coordinate is 0.
Plotting points is a crucial step in visualizing the relationship between coordinates and is the foundation for graphing lines accurately.

Slope Formula

To find the slope of the line passing through the points (2,3) and (4,0), we use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in our points, where \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (4, 0) \), we get:

• \[ m = \frac{0 - 3}{4 - 2} = \frac{-3}{2} \]
• The slope (m) tells us how steep the line is. A negative slope of \( -\frac{3}{2} \) means that for every 2 units we move to the right along the x-axis, the line descends 3 units downward.

This is key to understanding the direction and steepness of the line we are about to graph.

Graphing Lines

With the points (2,3) and (4,0) plotted and the slope calculated as \( -\frac{3}{2} \), we can now graph the line. Start by drawing a straight line through the plotted points. Remember, the line will fall 3 units for every 2 units you move to the right.
When graphing lines:

• Use a ruler to ensure the line is straight.
• Extend the line beyond your points for a complete graph.
• Label the points for clarity.

Graphing provides a visual representation of the function, helping you to better understand how changes in the x-values affect the y-values.