Problem 22 - Algebra 1: Concepts and Skills

Question

Find the slope and y-intercept of the graph of the equation. $$y=-2$$

Step-by-Step Solution

Answer

The slope of the graph is 0 and the y-intercept is -2.

Step 1: Identify the form of the equation

The equation \(y=-2\) is already in slope-intercept form, which is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept.

Step 2: Determine the slope

In a horizontal line like this, the slope \(m\) is 0, because the line does not rise or fall.

Step 3: Find the y-intercept

The y-intercept \(c\) is the value of \(y\) when \(x=0\). In this case, for all \(x\), \(y=-2\), so the y-intercept is -2.

Key Concepts

Slope

The slope of a line is an important concept when studying linear equations. The slope tells us how steep a line is, and whether it is increasing or decreasing as it moves from left to right. To find the slope, we usually look at the equation of the line in the form of the slope-intercept equation, which is written as:\[ y = mx + c \] Here, \( m \) represents the slope of the line. For the equation \( y = -2 \), this is actually the equation of a horizontal line. A horizontal line has a slope of 0 because it does not rise or fall as it moves along the graph. In other words, there is no change in the "vertical" value of \( y \) no matter how far you look to the "horizontal" on the \( x \)-axis. Horizontal lines are always parallel to the \( x \)-axis, and therefore have a zero slope. This characteristic makes it easy to identify horizontal slopes by just examining the equation of the line.

Y-Intercept

The y-intercept is another fundamental aspect of linear equations. It represents the point at which the line crosses or intercepts the \( y \)-axis. In the slope-intercept form \( y = mx + c \), the y-intercept is given by the constant \( c \).The y-intercept gives us a starting point on the \( y \)-axis when \( x \) is 0. For the equation \( y = -2 \), the y-intercept is straightforward to see.

• The line crosses the \( y \)-axis at \( -2 \).

This means when \( x \) is 0, \( y \) is equal to \( -2 \). This is the unique point where the line interacts with the \( y \)-axis, making it a key point of reference when graphing the equation. The y-intercept helps to plot the line quickly and is central to understanding where the line stands in relation to the origin point \( (0, 0) \).

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. The formula:\[ y = mx + c \]encapsulates both the slope and the y-intercept of the line. It is a simple yet powerful representation.

• \( m \) represents the slope of the line.
• \( c \) represents the y-intercept of the line.

This form makes it easy to graph a line, as you only need to know these two parameters: the slope and the y-intercept. For the equation \( y = -2 \), the slope-intercept form looks a little different: - Here, \( m = 0 \) and \( c = -2 \). - It simplifies the expression to just \( y = -2 \), a special case of the slope-intercept equation representing a horizontal line.Using the slope-intercept form, you can instantly see that the slope is 0, making the line horizontal, and the y-intercept is \( -2 \). This understanding allows you to easily plot lines and comprehend their orientation on a graph.