Problem 47 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Slope is 0; y-intercept is (0, 4).

Step 1: Identify the Equation Type

The given equation is in the form \(y = 4\), which is a horizontal line. In this form, \(y\) is constant for any value of \(x\).

Step 2: Determine the Slope

For a horizontal line in the form \(y = c\), where \(c\) is a constant, the slope is \(0\) because the line does not rise or fall as \(x\) changes.

Step 3: Identify the Y-Intercept

In the equation \(y = 4\), the line intersects the \(y\)-axis at \(y=4\). Therefore, the \(y\)-intercept is \((0, 4)\).

Step 4: Draw the Graph

To graph the line \(y = 4\), draw a horizontal line passing through the point \((0, 4)\). This line is parallel to the \(x\)-axis.

Key Concepts

Equation of a Line

An equation of a line is a mathematical expression that describes all the points along a straight path on a graph. One popular way to represent the equation of a line is through the slope-intercept form, given by the equation \( y = mx + b \). Here, \( m \) represents the slope of the line, indicating how steep the line is, while \( b \) represents the \( y \)-intercept, the point where the line crosses the \( y \)-axis.
For example, in the equation \( y = 4 \), there is no \( x \)-term. This indicates that the slope \( m \) is \( 0 \), resulting in a horizontal line with a constant \( y \)-value. The constant \( b = 4 \) specifies the point where the line crosses the \( y \)-axis. Therefore, the equation of this line clearly states that no matter what \( x \) is, \( y \) will always be 4.
Understanding the equation of a line in slope-intercept form helps in quickly graphing it and identifying important characteristics like its slope and \( y \)-intercept.

Horizontal Line

A horizontal line is one of the simplest forms of linear equations. It is characterized by having a constant \( y \)-value, meaning no matter how \( x \) changes, \( y \) remains the same. In mathematical terms, a horizontal line is represented as \( y = c \), where \( c \) is a constant. This constant is the \( y \)-value for all points on the line.
For instance, given the equation \( y = 4 \), we know it's a horizontal line because \( y \) is constant at 4 for any \( x \). The slope of a horizontal line is always zero because there is no rise or fall.
- **Slope:** 0 (no vertical change)- **Equation form:** \( y = c \)- **Graph:** Parallel to the \( x \)-axis

• All points on the line have the same \( y \)-coordinate
• Example: (0,4), (1,4), (-1,4)

Horizontal lines are particularly easy to graph, as they simply extend straight across the graph horizontally at the specified \( y \)-value.

Graphing Linear Equations

Graphing linear equations is a fundamental skill in mathematics, allowing you to visually represent the relationship between two variables. To graph a linear equation correctly, you often need the equation's slope and \( y \)-intercept.
Let's look at the equation \( y = 4 \). This is a special case because, as a horizontal line, it does not technically depend on the slope-intercept form. However, understanding how to approach graphing it is crucial.
**Steps to graph a linear equation like \( y = 4 \):**

• Identify the type of line: Recognize that \( y = 4 \) is a horizontal line, meaning the slope is zero.
• Locate the \( y \)-intercept: For \( y = 4 \), the \( y \)-intercept is at the point (0, 4).
• Plot the line: Draw a straight horizontal line passing through the point (0, 4). All points on this line will have \( y = 4 \).

This simple process allows you to visualize the equation on a graph, helping to reinforce the concept that a horizontal line shows no vertical change. It remains at a constant height across the \( x \)-axis, illustrating its consistent \( y \)-value.