Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The slope is 0 and the y-intercept is (0, 4), representing a horizontal line at \( y = 4 \).

Step 1: Identify the Equation Type

The given equation is in the form of a constant: \[ y = 4 \]This is a special linear equation that represents a horizontal line.

Step 2: Determine the Slope

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

Step 3: Determine the Y-Intercept

The y-intercept occurs where the line crosses the y-axis. Since the equation is \( y = 4 \), the line crosses the y-axis at 4. Thus, the y-intercept is the point \((0, 4)\).

Step 4: Sketch the Graph

To sketch the line \( y = 4 \):1. Plot the y-intercept at (0, 4).2. Draw a horizontal line across the graph that passes through y = 4, parallel to the x-axis.

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep or flat the line is. It's sometimes referred to as the "rise over run" because it tells you how much the line goes up or down (rise) for a certain amount it goes across (run). A positive slope slants upward, while a negative slope slants downward.
In the case of the equation \( y = 4 \), we have a horizontal line. This means it does not go up or down as it runs left or right.

• Horizontal lines always have a slope of zero.
• In mathematical terms, if the line doesn’t rise, the slope \( m = 0 \).
• This is because there is no change in the y-values, no matter the change in x.

A slope of zero indicates no incline, meaning the line is perfectly horizontal. Always remember: for horizontal lines like \( y = 4 \), the slope is \( m = 0 \).

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It is a fundamental part of a line's equation in the slope-intercept form \( y = mx + b \), where \( b \) represents the y-intercept.
For the equation \( y = 4 \), it is already in its simplest form, showing that it directly intersects the y-axis at \( y = 4 \).

• The y-intercept here is \((0, 4)\).
• This is the point on the graph where the line will meet the y-axis, in this instance directly at the number 4.
• In graphical terms, it’s the starting point of the line when drawing it on a Cartesian plane.

The y-intercept is especially important because it helps to define the starting position of the line on the graph.

Graphing Linear Equations

Graphing linear equations gives a visual representation of the relationships described by the equations. It involves plotting the y-intercept and using the slope to find other points on the line.
For the equation \( y = 4 \), this process is simple.

• Start by plotting the y-intercept, which is the point \((0, 4)\).
• Since the line is horizontal, no slope calculations are needed.
• Simply draw a straight, horizontal line that runs across the graph at \( y = 4 \), parallel to the x-axis.

In effect, graphing shows that every point on this line has a y-coordinate of 4. This confirms the nature of horizontal lines like those represented by equations of the form \( y = c \). Graphing is a powerful tool for understanding equations and visualizing relationships.