Problem 6

Question

In Exercises $5-8,$ let $L$ be the line determined by points $A$ and $B .$ $\begin{array}{ll}{\text { (a) Plot } A \text { and } B .} & {\text { (b) Find the slope of } L} \\ {\text { (c) Draw the graph of } L .}\end{array}$ $$A(-2,-1), \quad B(1,-2)$$

Step-by-Step Solution

Verified

Answer

The slope of the line $L$ defined by points $A(-2,-1)$ and $B(1, -2)$ is -1/3.

1Step 1: Plot Points A and B

Start by plotting the points $A(-2,-1)$ and $B(1, -2)$ on a 2-D plane. Point $A$ is located 2 units to the left of the y-axis and 1 unit below the x-axis. Point $B$ is located 1 unit to the right of the y-axis and 2 units below the x-axis.

2Step 2: Calculate the Slope

The slope $m$ of line $L$ is given by $m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$, where $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$. Substituting the coordinates of $A$ and $B$ into the equation gives $m = \frac{-2-(-1)}{1-(-2)} = -\frac{1}{3}$. So, the slope $m$ of the line $L$ is -1/3.

3Step 3: Draw the Graph

Draw line $L$ that passes through points $A$ and $B$, having the slope $m = -1/3$. The line will be going downwards as it moves from left to right since the slope is negative.

Key Concepts

Slope of a lineGraphing on Cartesian planePlotting points

Slope of a line

The slope of a line is a measure of its steepness. It shows how much the line rises or falls as you move along it horizontally. If a line goes upwards from left to right, it has a positive slope. If it goes downwards, the slope is negative. To find the slope ($ m $), we use the formula:

\[m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\]

Here, $ (x_{1}, y_{1}) $ and $ (x_{2}, y_{2}) $ are the coordinates of two points on the line. Let's look at points $ A(-2, -1) $ and $ B(1, -2) $. By plugging these into the formula, we get:

\[m = \frac{-2 - (-1)}{1 - (-2)} = -\frac{1}{3}\]

This calculation tells us that for every 3 units of horizontal movement to the right, the line drops by 1 unit. This is why its slope is $-1/3$.

Graphing on Cartesian plane

The Cartesian plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane corresponds to a pair of values (x, y), indicating horizontal and vertical distances from the origin $ (0, 0) $. To graph a line, you need at least two points. These points will help determine the line's specific position and direction.In our example, the points are $ A(-2, -1) $ and $ B(1, -2) $. Plot these points:

Move 2 units to the left and 1 unit down for point A.
Move 1 unit to the right and 2 units down for point B.

The next step is drawing a line through these plotted points. Because the slope $-1/3$ is negative, this line will slant downwards from left to right. The line reflects the movement indicated by the slope, connecting point A to point B, and continuing in both directions.

Plotting points

Plotting points is like locating places on a map, where each "address" is a pair of numbers known as coordinates. Each coordinate consists of an $ x $ value (how far left or right) and a $ y $ value (how far up or down).When plotting the point $ A(-2, -1) $: