Problem 4 - Precalculus Mathematics for Calculus

Question

(a) The slope of a horizontal line is _____. The equation of the horizontal line passing through \((2,3)\) is _____. (b) The slope of a vertical line is _____. The equation of the vertical line passing through \((2,3)\) is _____.

Step-by-Step Solution

Answer

(a) Slope is 0, equation is \( y = 3 \). (b) Slope is undefined, equation is \( x = 2 \).

Step 1: Understanding Horizontal Lines

A horizontal line is parallel to the x-axis. This means it does not rise or fall as you move along the x-axis.

Step 2: Finding the Slope of a Horizontal Line

For horizontal lines, the slope is zero because there is no vertical change as you move along the line.

Step 3: Writing the Equation of a Horizontal Line

The equation of a horizontal line is given by the form \[ y = c \]where \( c \) is the constant y-value throughout the line. Since the line passes through (2,3), its equation is \( y = 3 \).

Step 4: Understanding Vertical Lines

A vertical line is parallel to the y-axis. This means it has an undefined slope because it rises or falls infinitely along the y-axis.

Step 5: Finding the Slope of a Vertical Line

The slope of a vertical line is undefined because the change in \( x \) is zero, and division by zero is undefined.

Step 6: Writing the Equation of a Vertical Line

The equation of a vertical line is given by the form \[ x = c \]where \( c \) is the constant x-value throughout the line. Since the line passes through (2,3), its equation is \( x = 2 \).

Key Concepts

Horizontal Lines

Horizontal lines are lines that extend left to right across the coordinate plane, without any upward or downward movement, parallel to the x-axis. Understanding these lines is key:

• They have a constant y-value for all points on the line.
• The slope, or steepness, of these lines, is zero. This is because the change in vertical height is zero as you move along the line.

Once you understand that there is no vertical change, it becomes easy to grasp why the slope is zero. Mathematically, slope is defined as the change in y divided by the change in x, \[ m = \frac{\Delta y}{\Delta x} \]. For a horizontal line, \( \Delta y = 0 \), making \( m = 0 \).
Furthermore, the equation in the form \( y = c \), where \( c \) is the constant y-value, fully describes the line. For a line passing through the point \( (2,3) \), the equation is simply \( y = 3 \), suggesting that no matter what the x-value is, the y-value remains 3.

Vertical Lines

Vertical lines are lines that extend up and down, parallel to the y-axis, and are quite different compared to horizontal lines. Here’s what you need to know:

• These lines have a constant x-value for all points on the line.
• The slope of vertical lines is undefined. This happens because the horizontal distance, or change in x, is zero, making division by this zero impossible in mathematical terms.

To express the slope of a vertical line, consider \( m = \frac{\Delta y}{\Delta x} \). Here, \( \Delta x = 0 \), resulting in an undefined slope as division by zero is undefined.
The equation of such a line, where \( x \) remains constant, is given by \( x = c \). For a vertical line running through the point \( (2,3) \), the equation is simply \( x = 2 \). This means that no matter what the y-value is, the x-value stays fixed at 2 throughout.

Equations of Lines

The fundamental structure of line equations helps us describe their behavior and position on a graph. There are specific formats for horizontal and vertical lines, but in general, lines can be represented using different types of equations:

• Standard Form \[ Ax + By = C \]
• Slope-Intercept Form \[ y = mx + b \]

For horizontal lines, the equation \( y = c \) fits into the slope-intercept form where slope \( m = 0 \) and the line's position \( b \) is \( c \). Vertical lines, however, cannot be expressed in slope-intercept form due to their undefined slope.
Understanding the distinct equations that describe both types of lines helps significantly in plotting them on the coordinate plane. For any line, identifying points through which it passes and recognizing its slope are critical steps to forming its specific equation. This is especially true for solving problems related to line geometry and coordinate planes.