Problem 51 - Precalculus Mathematics for Calculus

Question

Find the slope and \(y\) -intercept of the line and draw its graph. $$y=4$$

Step-by-Step Solution

Answer

Slope: 0, y-intercept: 4. Graph is a horizontal line at y = 4.

Step 1: Identify the Equation Type

First, examine the given equation. The equation is given as \( y = 4 \). This is an equation of a horizontal line.

Step 2: Determine the Slope

For the equation of a horizontal line \( y = c \), where \( c \) is a constant, the slope \( m \) is always \( 0 \). Therefore, the slope of the line \( y = 4 \) is \( 0 \).

Step 3: Determine the y-intercept

The y-intercept of a line is the point where the line crosses the \( y \)-axis. For \( y = 4 \), the line crosses the \( y \)-axis at the point \( (0, 4) \). Thus, the y-intercept is \( 4 \).

Step 4: Graph the Line

A horizontal line with \( y = 4 \) is drawn parallel to the \( x \)-axis, passing through the point \( (0, 4) \) on the \( y \)-axis. It extends left and right infinitely.

Key Concepts

Slope

The slope of a line is a measure of its steepness and direction. It tells us how much the line inclines or declines as we move from left to right. For linear equations in the form of \( y = mx + b \), the coefficient \( m \) represents the slope. When the slope is positive, the line rises as it moves to the right. Conversely, a negative slope means the line falls as it moves to the right.

In the case of a horizontal line like \( y = 4 \), there is no vertical change no matter how far you travel along the \( x \)-axis. This means the rise is zero, resulting in a slope of zero.

• Horizontal Lines: Slope is 0
• Vertical Lines: Slope is undefined
• Positive Slope: Line rises from left to right
• Negative Slope: Line falls from left to right

Y-Intercept

The y-intercept is the point where the line crosses the \( y \)-axis. It is an important aspect of understanding the position of the line relative to the origin. This point can be represented as \( (0, b) \), where \( b \) is the y-coordinate when \( x \) is zero.

For the equation \( y = 4 \), the y-intercept is clearly at \( (0, 4) \). The entire line sits parallel to the \( x \)-axis, four units above it, maintaining a constant y-value of 4. Additionally, because this is a horizontal line, there is no way the x-value will affect the y-value. Understanding the y-intercept helps to picture the line's position without requiring graph plotting.

Graph of a Line

The graph of a line is a visual representation of all its solutions. A linear equation like \( y = 4 \) can be expressed graphically as a straight line. Here, this particular equation signifies a horizontal line parallel to the \( x \)-axis.

In drawing the graph of \( y = 4 \):

• Start at the y-intercept, which is point \( (0, 4) \).
• Since the slope is 0, draw a horizontal line through point \( (0, 4) \).
• The line spreads infinitely in both left and right directions across the plane.

This simplicity of horizontal lines such as \( y = 4 \) makes them easy to understand as they maintain a constant y-value across all points.