Problem 3 - College Algebra 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. $$(-2,1) \text { and }(2,2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (-2,1) and (2,2) is 1/4. Therefore, the line rises.

Step 1: Slope Calculation

Plug in the values of the coordinates into the slope formula. This gives: \(m = (2-1) / (2 - (-2)) = 1/4\). Even though the answer is a fraction, it is positive, implying that the line rises.

Step 2: Line Evaluation

Since the slope is positive, it can be concluded that the line rises. If the slope was negative, it would have meant the line falls. If the slope was undefined (which occurs when the denominator is zero), the line would be vertical. And if slope is zero, the line would be horizontal. Hence, in this case, the line rises.

Key Concepts

Coordinate Geometry

In coordinate geometry, we analyze the relationships between geometric figures and their positions on a coordinate plane. This branch of mathematics allows us to describe lines, shapes, and their properties using coordinates and algebraic equations.

For a line, we use pairs of points, each defined by a coordinate in the two-dimensional plane. A point such as \((-2, 1)\) indicates a position where the x-coordinate is \(-2\) and the y-coordinate is \(1\). By using coordinate geometry, we can determine various line characteristics, such as slope, intercepts, and angles formed with axes.

• Analyzing the positions of points helps to determine how lines behave and interact.
• Using a coordinate plane allows for precise calculations that describe geometric entities.

Understanding these foundational concepts is essential for further exploration of lines and slopes.

Slope Formula

The slope of a line is a number that describes how steep a line is and in which direction it inclines or declines. When given two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope formula is expressed as:
\[m = \frac{y_2-y_1}{x_2-x_1} \]
This formula measures the gradient or incline of the line by comparing the change in y-coordinates to the change in x-coordinates. If we plug in the given points \((-2,1)\) and \((2,2)\), we use the formula as follows:
\[m = \frac{2 - 1}{2 - (-2)} = \frac{1}{4} \]

• The numerator \(2 - 1\) shows the vertical change between the points.
• The denominator \(2 - (-2)\) represents the horizontal change.

By using this formula, we establish a quantitative way to interpret the line's behavior.

Positive Slope

When a line has a positive slope, it means the line inclines as it moves from left to right on the coordinate plane. This is indicated by a slope value greater than zero.

• A positive slope suggests the line "rises" upwards.
• In our example, since \( m = \frac{1}{4} \), the line rises as we go from left to right.

This characteristic is important because it allows us to visually interpret how a line behaves even before graphing it. When students understand positive slopes, they can easily categorize and predict the direction of lines in various problems.

Line Evaluation

Line evaluation involves interpreting the slope and understanding how the line appears in relation to the coordinate axes. Using our previous calculations, we know:

• If the slope is positive, the line rises.
• If the slope is negative, the line falls.
• A zero slope indicates a horizontal line.
• An undefined slope (caused by a zero in the denominator) indicates a vertical line.

In this exercise's case, the slope is \( \frac{1}{4} \). Since this slope is positive, we can confidently determine that the line rises.

Line evaluation helps students develop an intuition for geometric relationships and prepare them for more complex analyses with graphs and shapes.