Problem 33 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$y=-2$$

Step-by-Step Solution

Answer

The graph is a horizontal line at \( y = -2 \).

Step 1: Understanding the Equation

The equation given is \[ y = -2 \]This is a horizontal line where the y-value is always -2, regardless of the x-value.

Step 2: Finding Three Solutions

Since the equation is horizontal, we can choose any values for x and the y-value will always remain -2. Thus, three points that satisfy the equation are: 1. \( (0, -2) \)2. \( (1, -2) \)3. \( (-1, -2) \)

Step 3: Drawing the Graph

To draw the graph, plot the three points \( (0, -2), (1, -2), ext{ and } (-1, -2) \).Connect these points with a straight horizontal line extending in both directions along the x-axis. The line will be parallel to the x-axis, with all the y-coordinates being -2.

Key Concepts

Horizontal line

A horizontal line is one of the simplest types of graphs you will encounter in mathematics. It runs perfectly horizontally, parallel to the x-axis, and is proposed by an equation of the form \( y = k \), where \( k \) is a constant value. In the given problem, the equation \( y = -2 \) represents a horizontal line. This means that every point on this line has a y-coordinate of -2, while the x-coordinate can be any real number. Understanding horizontal lines is crucial as they graphically depict that a dependent variable (typically y) remains constant regardless of changes in the independent variable (x). Whether your x-value is 0, 50, or -99, the y-value will consistently be -2 in our example. The aesthetic simplicity of horizontal lines lies in their unchanging nature, making them predictable and easy to plot.

Coordinate points

Coordinates are essential in location-pointing and graphing on a two-dimensional plane, like the Cartesian plane you're dealing with. A coordinate point is in the form \( (x, y) \) and pinpoints a specific location on a graph. The first value in the pair, \( x \), represents the horizontal position, while the second value, \( y \), dictates the vertical position.For the exercise problem, the solution identifies three coordinate points: \( (0, -2) \), \( (1, -2) \), and \( (-1, -2) \).

• \( (0, -2) \) points directly two units below the origin on the y-axis.
• \( (1, -2) \) and \( (-1, -2) \) show the line's continuation left and right of the y-axis.

These coordinate points make it straightforward to visualize and construct the horizontal line on the graph. Simply plot the points and draw a line passing through them. Knowing how to use coordinate points effectively is fundamental to graphing any linear or non-linear functions.

Linear equations

Linear equations are algebraic expressions representing relationships between two variables. The general form is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. In the equation \( y = -2 \), which is a specific type of linear equation, the slope \( m \) is 0 because the line is horizontal, and the y-intercept \( b \) is -2. This implies that irrespective of the x-value, the output value for y remains constant at -2.Linear equations are foundational for understanding how to model relationships where one quantity depends on another. Whether you're measuring distance over time or mechanical prowess throughout motion, linear equations help define these interactions through their simple format. Graphically, these translate to a straight line on the graph, whether it's slanted, vertical, or horizontal like in our current investigation with \( y = -2 \). The coherence of linear equations provides both clarity and insights into basic and advanced mathematics.