Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-6: Sketch the line through \(A\) and \(B\), and find its slope \(m\). $$ A(4,-2), \quad B(-3,-2) $$

Step-by-Step Solution

Answer

The line is horizontal with a slope of 0.

Step 1: Identify the Coordinates

The problem provides two points with their coordinates: point \( A \) with coordinates (4, -2) and point \( B \) with coordinates (-3, -2). We will use these coordinates to sketch the line and find its slope.

Step 2: Plot the Points on a Graph

Draw a Cartesian coordinate system with an x-axis and y-axis. Locate point \(A\) at (4, -2), which means 4 units along the x-axis and -2 along the y-axis. Locate point \(B\) at (-3, -2), moving -3 units along the x-axis and staying the same vertically at -2 on the y-axis. Plot these points and draw a straight line through them.

Step 3: Use the Slope Formula

The formula for the slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the coordinates of points \(A\) and \(B\) into this formula.

Step 4: Substitute Coordinates into the Slope Formula

Substitute \((x_1, y_1) = (4, -2)\) and \((x_2, y_2) = (-3, -2)\) into the slope formula: \[ m = \frac{-2 - (-2)}{-3 - 4} \]

Step 5: Simplify the Expression

Simplify the expression to calculate the slope:\[ m = \frac{-2 + 2}{-3 - 4} = \frac{0}{-7} = 0 \] The slope \( m \) of the line through points \(A\) and \(B\) is 0, indicating a horizontal line.

Key Concepts

Coordinate Geometry

Coordinate Geometry is the branch of mathematics that connects algebra and geometry through graphs and coordinates. It allows us to visually analyze mathematical concepts using the Cartesian coordinate plane, which comprises two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Any point in this plane can be represented as an ordered pair

• The first number is the x-coordinate, showing the horizontal position.
• The second number is the y-coordinate, displaying the vertical position.

The power of coordinate geometry is its ability to handle problems involving figures like lines and curves by translating geometric properties into algebraic equations. This allows us to easily sketch graphs, calculate distances, and understand the relationship between points and lines.

Graphing Lines

Graphing lines is the process of drawing line segments between points in a coordinate plane. To graph a line, you need at least two points through which the line will pass. 1. **Plot the Points**: Begin by plotting your given points on the plane. Each point is located by its coordinates

• e.g., Point A(4, -2) is 4 units to the right of the origin and 2 units down.
• Point B(-3, -2) moves 3 units to the left of the origin and also 2 units downward.

2. **Draw the Line**: Once the points are plotted, use a straightedge to draw a line that passes through them. This represents all points along the straight path connecting them. Graphing lines helps visualize and solve problems related to linear equations or determine relationships between plotted points on a graph.

Horizontal Line

A horizontal line is a straight line with constant y-coordinates across all its points. In simple terms, it runs parallel to the x-axis and does not incline or decline. - When you have two points like A and B that share the same y-coordinate, the line formed is horizontal. - For example, both points A(4, -2) and B(-3, -2) have -2 as their y-coordinate. This implies that no vertical movement occurs between these points, hence forming a flat, horizontal line. Horizontal lines can be easily recognized thanks to their uniformity across the y-axis, making equations involving them simple.

Slope Formula

The slope formula is fundamental in coordinate geometry to determine the steepness or incline of a line. The slope of a line through two points labeled as y: So, for two points

• If you have two points (4, -2) and (-3, -2):
• The formula becomes:
• Solve for the numerator first -> -2 - (-2) = 0
• The denominator simplifies -> -3 - 4 = -7
• The slope is then 0 divided by -7, which equals 0.

A slope of 0 indicates a horizontal line, confirming the earlier observation about points A and B. The slope formula is a vital tool for understanding line behaviors and is crucial for defining relationships in graphs.