Problem 34 - Algebra 1

Question

Write the equation in slope-intercept form. Then graph the equation. $$ y=-2 $$

Step-by-Step Solution

Answer

The given equation is already in slope-intercept form with slope 0 and y-intercept -2. It represents a horizontal line crossing the y-axis at (0, -2).

Step 1: Recognize the equation's form

The given equation is \(y = -2\). This is already in slope-intercept form of a line, which is \(y = mx + b\) where m is the slope of the line and b is the y-intercept.

Step 2: Determine the slope and y-intercept

Since there's no x-term in the equation, the slope m is 0 and the y-intercept is -2. This means the line is horizontal and crosses the y-axis at the point (0,-2).

Step 3: Graph the equation

Sketch a horizontal line that passes through the y-axis at the point (0, -2). The line represents all the solutions to the equation.

Key Concepts

horizontal line

A horizontal line is one of the simplest types of lines you can encounter in graphing. This type of line runs parallel to the x-axis and has a unique characteristic: it has a slope of zero.
The equation of a horizontal line can often be written in the form of \(y = c\), where \(c\) is a constant value.
This means that no matter which point on the line you look at, the y-value remains the same. For example, in the equation \(y = -2\), every point on this line has a y-coordinate of -2, such as (1, -2), (-3, -2), and (5, -2).

• A horizontal line has an undefined x-intercept, except in a rare case where it lies along the x-axis.
• The slope (\[m = 0\]) indicates it's neither rising nor falling.
• It's always parallel to the x-axis regardless of its y-value.

Recognizing a horizontal line from its equation is straightforward when the equation lacks an x-term.

graphing linear equations

Graphing linear equations is an essential skill in algebra, helping you visualize the relationships between variables.
There are different forms in which a linear equation can appear, but a common one is the slope-intercept form, \(y = mx + b\). Here, \(m\) represents the slope, while \(b\) is the y-intercept.
When graphing such an equation, begin by plotting the y-intercept on the y-axis. This point is where the line crosses the y-axis, and its coordinates are (0, b).

• For the equation \(y = -2\), the graph starts at (0, -2).
• Use the slope, \(m\), to find additional points. However, with a horizontal line like \(y = -2\), the slope is 0, leading to no vertical change as you move along the x-axis.
• Draw the line through these points, maintaining its parallel alignment with the x-axis.

Graphing provides a visual representation of all possible solutions to the equation, determined by infinite points along the line.

y-intercept

The y-intercept of a line is the point at which the line crosses the y-axis. It is a crucial element in the slope-intercept form of a linear equation, \(y = mx + b\), where \(b\) denotes the y-intercept.
For the equation \(y = -2\), the y-intercept is clearly \(-2\), showing that the graph intersects the y-axis at the point (0, -2).

• The y-intercept is significant, especially for evaluating and sketching the graph of the equation.
• It gives a definite starting point when drawing the line on a coordinate plane.
• Identifying the y-intercept helps understand the vertical position of the line in relation to the origin.
• A change in the y-intercept shifts the entire line up or down, without altering its slope.

Recognizing the y-intercept helps solidify the understanding of linear equations and their graphical depictions.