Problem 65 - Algebra A Combined Function

Question

Use your own graph paper to draw a line parallel to the line \(y=-1\) that intersects the \(y\) -axis at -4 . What is the equation of this line?

Step-by-Step Solution

Answer

The equation of the line is \(y = -4\).

Step 1: Understand the Given Line

The given line is described by the equation \(y = -1\). This line is a horizontal line, meaning it has no slope and all points on this line have a \(y\)-coordinate of \(-1\).

Step 2: Determine Characteristics of the New Line

The task requires drawing a line parallel to the line \(y = -1\). Parallel lines have the same slope, and since \(y = -1\) is horizontal, the new line must also be horizontal. Therefore, it will have the same form: \(y = c\), where \(c\) is a constant.

Step 3: Find the Specific \(y\)-Intercept

The problem specifies that the line passes through the \(y\)-axis at \(-4\). Therefore, the \(y\) value of our new line, \(c\), must be \(-4\).

Step 4: Write the Equation of the New Line

Finally, using the characteristics of a horizontal line parallel to \(y = -1\) and the \(y\)-intercept provided, we write the equation of the line as \(y = -4\).

Key Concepts

Graphing Lines

Graphing lines on a coordinate plane involves plotting points that satisfy a linear equation and connecting these points to form a straight line. To graph a line, you typically need to identify its slope and y-intercept from its equation. The general equation of a line is given by the slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

• A positive slope means the line ascends from left to right.
• A negative slope means the line descends from left to right.
• A zero slope implies a horizontal line.

Once the slope and y-intercept are identified, you can plot the y-intercept on the vertical (y) axis, then use the slope to determine other points on the line by moving vertically and horizontally according to the rise over run principle. Finally, connect these points to draw the full line. Understanding how to graph lines is fundamental to visualizing equations in algebra.

Horizontal Line Equations

Horizontal lines are special because they have a slope of zero. This means they run parallel to the x-axis, maintaining a constant y-value along their entire length. The equations of horizontal lines are simple, typically in the form \(y = c\), where \(c\) is a constant. This constant represents the y-coordinate where the line intersects the y-axis. Key characteristics of horizontal lines include:

• The slope is always zero, indicating no rise or fall as the line progresses from left to right.
• They have no x-intercept unless they coincide with the x-axis (i.e., when \(c = 0\)).
• They are parallel to each other if they share the same slope, which for horizontal lines, is always zero.

Understanding horizontal lines is crucial for recognizing how they behave on the graph and serves as a basis for more complex graphing tasks.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form of a line's equation, \(y = mx + b\), the y-intercept is \(b\). For horizontal lines, the y-intercept is not just a point, but also dictates the equation of the line itself, \(y = c\), where it is crucial in determining the line's position.

• The y-intercept is where the x-coordinate is always zero, meaning it is the value of \(y\) when \(x\) is zero.
• Knowing the y-intercept helps in quickly sketching the line on a graph, as it is the starting point of the line.
• It is particularly easy to identify for horizontal lines, as all points on these lines have the same y-coordinate as the y-intercept.

Understanding the y-intercept is essential for equation manipulation and graph interpretation, making it a foundational concept in graphing and algebraic equations. It helps bridge the theoretical aspects of equations with their practical application on a graph.