Problem 41 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(y=-2\)

Step-by-Step Solution

Answer

Y-intercept is (0, -2); no x-intercept.

Step 1: Identify the Equation Type

The given equation is a horizontal line, as it does not have an "x" term and the "y" value is constant, which is given as \(y = -2\).

Step 2: Find the Y-Intercept

The equation \(y = -2\) directly shows the y-coordinate of the line. The y-intercept is the point where the line crosses the y-axis. For this equation, the y-intercept is \((0, -2)\).

Step 3: Find the X-Intercept

For any horizontal line, the x-values change along the line, but the line never intersects the x-axis unless \(y = 0\). Since \(y = -2\), the line does not cross the x-axis and hence there is no x-intercept.

Step 4: Graph the Equation

Since \(y = -2\) is a horizontal line, you draw a line parallel to the x-axis that goes through all points where y equals -2. The line will span horizontally and go through the point \((0, -2)\) on the graph.

Key Concepts

y-intercept

The y-intercept of a line is the point where it crosses the y-axis. In the equation form, it is usually represented by the point

• (0, b) in the equation y = mx + b, where b is the y-intercept.

When the given equation is

• y = -2, it means the y-intercept is directly given. Here, the line intercepts the y-axis at the point (0, -2).

In practical terms, if you were to draw this line on a graph, you would start your drawing at this intercept point on the y-axis. Understanding the y-intercept is key to starting off your graph correctly. It acts as a foundation to understanding where a line begins in terms of the vertical axis, regardless of the slope or other components of the equation.

horizontal line

A horizontal line is distinctive in its simplicity. It is parallel to the x-axis and does not incline upwards or downwards. This line is formed by an equation in the format of

• y = c, where c is a constant.

For example, in the equation

• y = -2, the entire line lies at y = -2,

meaning every point on this line has a y-coordinate of -2. There are no changes in the y-coordinate along the horizontal line, which is why it stretches left and right consistently. One of the defining characteristics of these lines is that they don't have an x-intercept unless they sit exactly on top of the x-axis itself, which would mean the equation is

• y = 0.

The absence of an x-intercept for most horizontal lines is important to remember because it hints at the line's constant y-value.

graphing equations

Graphing equations is a visual way to represent linear and other types of equations. It helps to illustrate the relationship between variables in a tangible way. For a line represented by the equation y = -2, the graphing process involves plotting this line on a coordinate plane. To graph the equation:

• Start by locating the y-intercept on the graph at (0, -2).
• Once the y-intercept is marked, draw a straight, horizontal line through this point.

This line will extend indefinitely from the left to the right, parallel to the x-axis. Remember, graphing isn't just a plotting exercise; it's about understanding the relation showcased via the line's direction and position on the grid.

linear equations

Linear equations are fundamental tools in algebra, often expressed in the form

• y = mx + b.

They represent straight lines on a graph, where m stands for the slope, and b refers to the y-intercept. However, sometimes linear equations skip the x-variable, as observed in equations like

y = -2.

These simpler forms just depict horizontal or vertical lines. The simplicity of linear equations makes them not only easy to graph but also predictable given their consistent rate of change. Equations without an x-term (or with a zero slope) like y = -2 show uniformity across their span. Recognizing these forms helps build a foundational understanding that can be applied to more complex functions and applications in mathematics.