Problem 20 - Algebra A Combined Function

Question

Write an equation of each line. See Examples 3 and \(4 .\) Slope \(0 ;\) through (-2,-4)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Slope

The slope of the line is given as 0. A slope of 0 indicates a horizontal line. Hence, the equation of the line will be of the form \(y = c\) for some constant \(c.\)

Step 2: Identify the Constant

Since the line passes through the point \((-2, -4)\) and is horizontal, every point on this line has the same \(y\)-value. Therefore, the constant \(c = -4.\)

Step 3: Write the Equation

Now that we know the line is horizontal and passes through \(y = -4\), the equation of the line is \(y = -4.\)

Key Concepts

Understanding Slope

In mathematics, the concept of the slope is crucial when discussing linear equations. The slope measures how steep a line is and the direction in which it tilts. To calculate the slope, we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

where \( m \) stands for slope, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on a line.
A positive slope means the line rises as it moves from left to right, while a negative slope indicates a fall.
When the slope is zero, the line is perfectly horizontal, meaning there is no rise or fall regardless of changes in the x-value. In our exercise, the slope is 0, signaling a horizontal line.

Horizontal Line

A horizontal line is a straight line that goes from left to right or right to left, remaining parallel to the x-axis.
Every point on a horizontal line has the same y-coordinate. This characteristic means the slope of a horizontal line is always zero.
When writing the equation of a horizontal line, it takes the form:

• \( y = c \)

where \( c \) is a constant that represents the y-coordinate of every point on the line.
In the given problem, since the line passes through the point \((-2, -4)\), every other point on the line also has a y-coordinate of \(-4\). Therefore, the equation is \( y = -4 \).

Graphing Linear Equations

Graphing linear equations helps visualize the relationship between variables. For linear equations, graphs are straight lines, characterized by constant slopes.
To graph a line, you first need two points. Once the slope is known, a single point is enough to draw the line since the slope tells you how to move from that point.

• For a slope of 0, the line is horizontal, staying at the same y-level across all x-values.
• The equation describes the line, like \( y = -4 \), which means every point \((x, y)\) on this line satisfies \( y = -4 \).

When graphing this in a coordinate plane, draw a straight line parallel to the x-axis at the height \(-4\). It shows all points where the y-value is \(-4\), irrespective of the x-value.