Problem 5 - Precalculus Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) $$P(0,0), Q(4,2)$$

Step-by-Step Solution

Answer

The slope of the line is \(\frac{1}{2}\).

Step 1: Understand the Slope Formula

The slope of a line through two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This represents the change in the y-values divided by the change in the x-values.

Step 2: Identify the Coordinates

Identify the coordinates for points \(P\) and \(Q\):- \(P(0,0)\) implies \(x_1 = 0\) and \(y_1 = 0\).- \(Q(4,2)\) implies \(x_2 = 4\) and \(y_2 = 2\).

Step 3: Plug in the Coordinates

Using the slope formula, substitute \(x_1, y_1, x_2,\) and \(y_2\) with their respective values:\[ m = \frac{2 - 0}{4 - 0} \]

Step 4: Simplify the Expression

Simplify the expression to find the slope:\[ m = \frac{2}{4} = \frac{1}{2} \]

Step 5: Conclusion

The slope of the line that passes through points \(P(0,0)\) and \(Q(4,2)\) is \frac{1}{2}\.

Key Concepts

Two-Point Formula

The two-point formula is a crucial concept when you want to find the slope of a line passing through two given points. Think of it as a simple formula that helps you measure how steep a line is. To compute the slope, you use the coordinates of two points, typically noted as

• Point 1: \((x_1, y_1)\)
• Point 2: \((x_2, y_2)\)

To find the slope, you plug these coordinates into the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the slope \(m\) by taking the difference in the \(y\)-coordinates and dividing it by the difference in the \(x\)-coordinates. Always remember that the change in \(y\) is on top (numerator) and change in \(x\) is on the bottom (denominator). This formula tells you how much a line rises or falls for each unit it moves horizontally.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry. It uses the coordinate plane to explore algebraic equations and geometric reasoning. In this coordinate system, points are represented by pairs of numbers \((x, y)\), which denote their positions on a graph.
The coordinate plane is divided into four quadrants, which helps in identifying and plotting points. Deducing relationships between points helps in forming linear equations and solving geometric problems.
Coordinate geometry is immensely helpful because it allows complex geometric shapes and lines to be expressed using simple algebraic equations. Knowing this makes it easier to solve problems involving distances, midpoints, and slopes of lines.

Linear Equations

Linear equations form the cornerstone of algebra and coordinate geometry. A linear equation forms a straight line when graphed on a coordinate plane, symbolizing a constant rate of change. Its general form is expressed as:

• \(y = mx + b\)

Here, \(m\) represents the slope, indicating the line's steepness, while \(b\) is the y-intercept, the point crossing the y-axis.
In practical scenarios, these equations are powerful tools used to describe real-world relationships between two variables. Whether determining profit with respect to production costs or distance over time, linear equations provide clear and visual solutions. Grasping the concept of a linear equation helps in exploring and understanding various mathematical and real-life phenomena.

Rate of Change

Rate of change is a broad concept that describes how one quantity changes relative to another. In the context of a straight line on a graph, it is simply the slope.
The slope, or rate of change, determines how fast something is moving or changing. For instance, when you see the slope as \(\frac{1}{2}\), it means that for every 2 units you move horizontally, the line moves 1 unit vertically.
A higher slope indicates a steeper line, suggesting rapid change, whereas a smaller slope indicates a gradual change. Understanding rate of change enables you to interpret the behavior of relationships shown on graphs effectively. It is widely used in various fields such as physics for velocity, economics for growth rates, and even biology to measure population changes.