Problem 4 - Precalculus Essentials

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(-1,3) \text { and }(2,4)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (-1,3) and (2,4) is \(\frac{1}{3}\) and the line rises.

Step 1: Identify the Coordinates of Both Points

The first given point is (-1,3) and the second point is (2,4). Therefore, \(x_1 = -1, y_1 = 3, x_2 = 2, y_2 = 4\).

Step 2: Calculate the Slope

This calculation can be made by plugging these values into the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Thus \( m = \frac{4 - 3}{2 - (-1)} = \frac{1}{3} \).

Step 3: Determine the Line's Direction

The slope is positive which means the line rises.

Key Concepts

Understanding Points and Coordinates

When you are working with points and coordinates in the context of a line, it's essential to understand what these terms mean. Each point consists of an ordered pair, \(x, y\), where \(x\) represents the horizontal position, and \(y\) represents the vertical position on a graph. These coordinates help locate the exact position of a point in a two-dimensional space.

In the exercise, we were given the points \((-1, 3)\) and \(2, 4\). This notation tells us that the first point has an \(x\)-coordinate of -1 and a \(y\)-coordinate of 3, while the second point has an \(x\)-coordinate of 2 and a \(y\)-coordinate of 4. By identifying these coordinates, we set the stage for calculating the line's slope. This is because the slope formula requires the substitution of these specific values.

What is a Positive Slope?

The concept of a positive slope is tied to how a line moves across a graph. A slope indicates how steep a line is and in which direction it goes. On a graph, if a line rises from left to right, it has a positive slope.

To find the slope between two points, we use the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula subtracts the \(y\)-coordinates and \(x\)-coordinates of the two points:

1. The difference in the \(y\)-coordinates (rise) tells us how much the line goes up or down
2. The difference in the \(x\)-coordinates (run) tells us how much the line goes left or right
For our points \((-1, 3)\) and \(2, 4)\), the slope is calculated as \(\frac{1}{3}\). The positive slope of \(\frac{1}{3}\) indicates the line rises gently from left to right.

Exploring Line Direction

Determining the line's direction helps in visualizing how the line behaves on a graph. After calculating the slope, understanding the direction becomes straightforward. A few simple rules can guide you:

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope means a vertical line.

In our exercise, with a slope of \(\frac{1}{3}\), the line clearly rises. This means as you move along the graph from left to right, the line will ascend upward. By grasping these key ideas, you can easily determine how any line behaves just by knowing its slope value.