Problem 9 - Precalculus: Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) \(P(-1,2), Q(0,0)\)

Step-by-Step Solution

Answer

The slope of the line is -2.

Step 1: Identify Coordinates of Points

The coordinates of the points are given as \( P(-1, 2) \) and \( Q(0, 0) \). Here, \( P \) is the first point with \( x_1 = -1 \) and \( y_1 = 2 \), and \( Q \) is the second point with \( x_2 = 0 \) and \( y_2 = 0 \).

Step 2: Recall the Slope Formula

The formula to calculate the slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 3: Substitute Coordinates into the Slope Formula

Using the coordinates identified, substitute into the formula: \( m = \frac{0 - 2}{0 - (-1)} \).

Step 4: Perform Calculations

Calculate the differences: \( y_2 - y_1 = 0 - 2 = -2 \) and \( x_2 - x_1 = 0 - (-1) = 1 \). So the slope \( m \) becomes \( m = \frac{-2}{1} = -2 \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric concepts. It deals with graphs and shapes on the coordinate plane.
By linking algebra and geometry, you can find more tangible solutions to problems.

Imagine plotting points on a grid, similar to battleship but with graphs and lines.

• Each point has an "address" using coordinates \( (x, y) \).
• Coordinates describe a point's position in two-dimensional space.

One of the fundamental parts of coordinate geometry is understanding how lines and points interact.
This involves calculating slopes, determining intersections, and finding distances.

Understanding coordinate geometry is essential for solving problems like finding slopes, which is a common requirement in analyzing lines on graphs.

Slope Formula

The slope formula is a mathematical tool used to measure the steepness or incline of a line in the coordinate plane.
It's crucial to understand the slope to explain how two points on a graph are connected by a straight line.

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

The numerator \( (y_2 - y_1) \) represents the change in the vertical direction (rise), while the denominator \( (x_2 - x_1) \) represents the change in the horizontal direction (run).

Depending on the sign of the result:

• A positive slope indicates the line ascends from left to right.
• A negative slope indicates the line descends from left to right.
• A zero slope means the line is horizontal.
• An undefined slope means the line is vertical.

Mastering this formula is necessary for analyzing the orientation and direction of lines.

Calculating Slope

Let's delve into how to calculate the slope using an example with given points.
For the points \(P(-1,2)\) and \(Q(0,0)\), follow the steps to find the slope of the line through them.

**Steps to calculate the slope:**1. **Identify coordinates**: Recognize the coordinates of each point on the graph. Here, \( x_1 = -1, y_1 = 2, x_2 = 0, y_2 = 0 \).2. **Apply the slope formula**: Use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).3. **Substitute values**: Put the values into the formula \( m = \frac{0-2}{0-(-1)} \).4. **Perform calculations**:

• Calculate the numerator: \( y_2 - y_1 = 0 - 2 = -2 \).
• Calculate the denominator: \( x_2 - x_1 = 0 - (-1) = 1 \).

5. **Determine the slope**: Your final step is to simplify: \( m = \frac{-2}{1} = -2 \).
By going through these steps, you can clearly understand and compute the slope between any two points.