Question

In Problems \(23-28\), find the slope of the line containing the given two points. (2,-4) \text { and }(0,-6)

Short answer

The slope of the line is 1.

Step-by-step solution

Understanding the slope formula

The formula to find the slope, \( m \), of a line that passes through two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula represents the change in \( y \) (vertical change) over the change in \( x \) (horizontal change).

Identify the given points

We are given two points: \( (2, -4) \) and \( (0, -6) \). Let's assign the coordinates: \( x_1 = 2, y_1 = -4, x_2 = 0, y_2 = -6 \).

Calculate the difference in y-values

Subtract the \( y \)-value of the second point from the \( y \)-value of the first point: \( y_2 - y_1 = -6 - (-4) = -6 + 4 = -2 \).

Calculate the difference in x-values

Subtract the \( x \)-value of the second point from the \( x \)-value of the first point: \( x_2 - x_1 = 0 - 2 = -2 \).

Compute the slope

Use the differences calculated in previous steps to compute the slope. Substitute them into the slope formula: \( m = \frac{-2}{-2} = 1 \).

Key concepts

Slope Formula

The slope formula is a fundamental concept in coordinate geometry that helps us determine how steep a line is. It's represented as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula is used to calculate the slope, \( m \), which measures the angle or inclination of a line between two points given their coordinates.

• It's important for understanding linear relationships in mathematics.
• Different slopes indicate different types of lines: positive slopes suggest an upward trend, while negative slopes indicate a downward trend.

Understanding the slope formula is crucial as it not only helps in graphing lines but also in interpreting data trends. It allows students to quickly perceive changes in direction and steepness between two points on a plane.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, combines algebra and geometry. It uses an algebraic approach to solve geometric problems by assigning coordinates.

In coordinate geometry, every point on a plane is defined by a pair of numbers known as coordinates: \((x, y)\). Each point is like an address where a specific line or curve intersects the axes. This system greatly simplifies many complex geometrical concepts and calculations.

• Coordinate geometry is used in various fields like physics for motion tracking and geography for mapping locations.
• It lays the foundation for calculus and other advanced mathematical concepts.

By using coordinate geometry, students can solve a variety of problems ranging from calculating distances to finding intersections of lines.

Horizontal Change

Horizontal change refers to the change or difference in the \(x\)-coordinates of two points on a graph. In simpler terms, it reflects how much we move along the x-axis.

For two points \((x_1, y_1)\) and \((x_2, y_2)\), the horizontal change is calculated as \(x_2 - x_1\).

• This difference indicates the distance between the points horizontally.
• It's vital for calculating the slope as the denominator of the slope formula.

Recognizing horizontal change helps students in visualizing and understanding shifts and translations in graphs, which is fundamental when analyzing functions and their behavior.

Vertical Change

Vertical change represents the shift or change in the \(y\)-coordinates between two points on a graph. Essentially, it shows how much we move up or down along the y-axis.

This is determined by finding the difference \(y_2 - y_1\) for points \((x_1, y_1)\) and \((x_2, y_2)\).

• Vertical change is the numerator in the slope formula and crucial for its calculation.
• It aids in understanding the rise or fall achieved when moving from one point to another.

Understanding vertical change is essential for students to see the complete picture when studying slopes, as it depicts the line's steepness and direction shift.