Problem 64 - Calculus Volume 1

Question

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. $$ (1,9) \text { and }(-8,5) $$

Step-by-Step Solution

Answer

The slope is \(\frac{4}{9}\), and the line is increasing.

Step 1: Identify Coordinates

First, identify the coordinates of the two points given in the problem. The points are \((x_1, y_1) = (1, 9)\) and \((x_2, y_2) = (-8, 5)\).

Step 2: Use the Slope Formula

To find the slope \(m\) of the line passing through the points, use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the values from the coordinates: \[ m = \frac{5 - 9}{-8 - 1} \]

Step 3: Simplify the Slope Fraction

Simplify the expression for the slope: \[ m = \frac{-4}{-9} = \frac{4}{9} \]So, the slope \(m\) is \(\frac{4}{9}\).

Step 4: Determine the Line's Behavior

Since the slope \(\frac{4}{9}\) is positive, the line is increasing. An increasing line means as \(x\) increases, \(y\) also increases.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where algebra is used to study geometric problems. It involves the use of a coordinate plane to represent geometric figures and analyze their properties. In a coordinate plane, each point is uniquely represented by a pair of numbers, called coordinates. These coordinates denote the position of the point in relation to two perpendicular number lines, usually called axes.

In our example, we are given two points: \((1,9)\) and \((-8,5)\). Here:

• The first number in each pair is the x-coordinate, which tells us how far along the horizontal axis the point is.
• The second number is the y-coordinate, representing the point's position along the vertical axis.

Understanding how to plot these points and calculate distances between them is essential in coordinate geometry. It allows you to visualize and analyze the properties of lines that pass through these points.

Line Equations

A line equation describes all the points along a straight line on the coordinate plane. The most common form of a line equation is the slope-intercept form: \[ y = mx + c \]where:

• \(m\) represents the slope of the line, indicating its steepness and direction.
• \(c\) is the y-intercept, the point where the line crosses the y-axis.

The equation of a line can also be derived using the point-slope form:\[ y - y_1 = m(x - x_1) \]This form is convenient when you already know a point on the line, \((x_1, y_1)\), and the slope, \(m\).

With the slope calculated for our line as \(\frac{4}{9}\), you could further explore how this line would look on a graph. Begin by identifying any specific points, such as the y-intercept, which requires solving for when \(x = 0\). These calculations turn abstract numbers into a concrete line on the plane, allowing easy predictions about other points the line would intersect.

Increasing and Decreasing Functions

The slope of a line tells us whether the function is increasing, decreasing, or constant. Here's what different slopes imply:

• A positive slope, like our \(\frac{4}{9}\), indicates an increasing function. This means as the x-values increase, the y-values also increase, giving the line an upward direction from left to right.
• A negative slope indicates a decreasing function, where an increase in x results in a decrease in y, making the line slope downwards from left to right.
• If the slope is zero, the line is horizontal, meaning the function is constant – y-values stay the same as x-values change.
• A vertical line, which corresponds to an undefined slope, implies no true functional relationship can be expressed between x and y, as it would defy the definition of a function (where each x has only one y corresponding to it).

Understanding these concepts helps in determining the behavior of a line based solely on its slope, allowing you to quickly assess the direction and nature of a given linear function.