Problem 17 - Algebra 1: Concepts and Skills

Question

Plot the points and draw a line that passes through them. Use the rise and run to find the slope. $$ (-2,1) \text { and }(2,4) $$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (-2,1) and (2,4) is 3/4.

Step 1: Plot the Points

Start by plotting the two points (-2,1) and (2,4) on a graph. Recall that the first coordinate of the point represents the position on the x-axis and the second coordinate represents the position on the y-axis. Each point is drawn as a dot on grid where the x-coordinate and y-coordinate meet.

Step 2: Draw the Line

Now, draw a straight line that passes through the two points (-2,1) and (2,4). Make sure the line extends past the points to accurately show the slope.

Step 3: Find the Slope

To find the slope of the line, apply the formula for the slope which is \(slope = (y2 - y1) / (x2 - x1)\). Here, (x1, y1) and (x2, y2) are the coordinates of the two points. So, substituting the given values, we get \(slope = (4 - 1) / (2 - (-2)) = 3 / 4\). Hence, the slope of the line through the points (-2,1) and (2,4) is 3/4.

Key Concepts

Plotting Points on a Graph

When working with coordinate geometry, plotting points on a graph is a fundamental skill. Each point has two coordinates, the x-coordinate and the y-coordinate. The x-coordinate tells us how far along the horizontal x-axis the point is, while the y-coordinate tells us how far along the vertical y-axis the point is.

To plot the points, you find the x-coordinate on the x-axis and move vertically to the level of the y-coordinate, and do the same for the y-axis but move horizontally. The intersection of these movements is where the point is plotted. For example, to plot (-2, 1), locate -2 on the x-axis, then move upwards to 1 on the y-axis and place a dot here. Repeat the same process for the point (2, 4).

Plotting points accurately is crucial for representing the problem visually, and leads to a better understanding of concepts like slope and the behavior of linear equations on a graph.

Slope Formula

Understanding the slope of a line is important as it measures the steepness and the direction of the line. The slope is calculated using two points on the line, referred to as \( (x1, y1) \) and \( (x2, y2) \).

The formula for slope is:
\[ slope = \frac{y2 - y1}{x2 - x1} \]
It represents the ratio of the vertical change (rise) to the horizontal change (run) between two points. If the slope is positive, the line slants upward to the right, and if it is negative, the line slants downward to the right.

Using the exercise's points (-2, 1) and (2, 4), we calculate the slope as follows:
\[ slope = \frac{4 - 1}{2 - (-2)} = \frac{3}{4} \]
The slope of 3/4 indicates that for every 3 units rise (upward movement), there is a 4 units run (rightward movement). This concept is central to understanding how different points relate and define the shape and direction of the line.

Graphing Linear Equations

Graphing linear equations involves drawing a line on a graph that represents all the solutions to the equation. A linear equation can be in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis.

To graph a linear equation, like the line passing through the points (-2, 1) and (2, 4) from our exercise, we can use the slope-intercept form. Alternatively, we can plot the points and draw the line that connects them, as done in the exercise.

Graphing these lines is not only about drawing; it's an exploration into the relationship that exists between the x and y variables. The line's slope tells us how y changes with x, meaning the rate of change from one point to another. Graphing linear equations is foundational to many fields, such as science, engineering, and economics, because it helps us visualize and interpret relationships between variables.