Problem 6 - Precalculus Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) $$P(0,0), Q(2,-6)$$

Step-by-Step Solution

Answer

The slope of the line through points \( P \) and \( Q \) is \( -3 \).

Step 1: Find Change in Y

First, identify the y-coordinates of the given points. For point \(P\), the y-coordinate is \(0\), and for point \(Q\), the y-coordinate is \(-6\). Calculate the change in y, \( \Delta y = y_2 - y_1 = -6 - 0 = -6 \).

Step 2: Find Change in X

Next, identify the x-coordinates of the given points. For point \(P\), the x-coordinate is \(0\), and for point \(Q\), the x-coordinate is \(2\). Calculate the change in x, \( \Delta x = x_2 - x_1 = 2 - 0 = 2 \).

Step 3: Calculate Slope

The slope of the line is given by the formula \( m = \frac{\Delta y}{\Delta x} \). Substitute the values from the previous steps: \( m = \frac{-6}{2} = -3 \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through the use of a coordinate system. This branch of mathematics merges algebra and geometry, allowing us to study geometry using an ordered pair of numbers or coordinates. By plotting points, lines, and curves on the Cartesian plane, coordinate geometry provides a way to describe a position in space.

This is hugely helpful in solving geometry problems, as it translates them into algebraic expressions. The points, like on our exercise, are usually denoted by coordinates in the form of \((x, y)\). The grid used in coordinate geometry is the coordinate plane, typically characterized by two numbered lines called axes.

• The horizontal axis, the x-axis
• The vertical axis, the y-axis

Each point on a plane is represented by its horizontal (x) and vertical (y) distances from these axes.

In our problem, the points are given as \(P(0,0)\) and \(Q(2,-6)\). These coordinates plot the position of each point regarding its distance from the origin \((0, 0)\). This form of geometry allows us to explore the relationships between angles, lengths, areas, and other properties of shapes. It becomes especially powerful when dealing with linear equations and slopes of lines.

Change in Y and X

When trying to find the slope of a line passing through two points, we make use of the concept of 'change in Y and X'. This involves calculating how much the vertical position (y-coordinate) changes as you move horizontally (x-coordinate) between two points.

In the formula for slope, represented as \(m\), you see we've calculated the change in y \((\Delta y)\) as the difference between the y-coordinates of two points:
\[ \Delta y = y_2 - y_1 \]
Similarly, the change in x \((\Delta x)\) is derived from the difference between x-coordinates:
\[ \Delta x = x_2 - x_1 \]

• For point \(P\), \((x_1, y_1) = (0, 0)\)
• For point \(Q\), \((x_2, y_2) = (2, -6)\)

So here, we calculate
\(\Delta y = -6 - 0 = -6\) and \(\Delta x = 2 - 0 = 2\).

This change, once located, helps us determine the slope. Keep in mind that change in these coordinates also visually represents the movement necessary to navigate from one point to another on a graph.

Linear Equations

A linear equation is an algebraic equation that represents a straight line on a graph. The foundational form of a linear equation is \(y = mx + c\), where \(m\) represents the slope and \(c\) represents the y-intercept. This equation shows the direct relationship between two variables, often x and y.

In terms of slopes, the slope tells us the steepness and direction of a line. It's calculated as \(m = \frac{\Delta y}{\Delta x}\), showcasing the vertical change per unit of horizontal change. The magnitude of the slope shows how steep a line is, while the sign indicates its direction: positive slopes rise from left to right and negative slopes fall.

• A slope of 0 indicates a perfectly horizontal line
• An undefined slope suggests a vertical line
• Our slope calculation from the problem yields -3, indicating a downwards slope

The reasoning behind linear equations is that they provide valuable insights into real-world problems, enabling complex relationships to be understood via simple straight-line formulas.

Whether analyzing cars' speed over time or determining profits against the number of items sold, linear equations serve as a crucial mathematical tool to describe patterns in a straightforward manner.