Problem 34 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$y=3$$

Step-by-Step Solution

Answer

Plot points (0, 3), (1, 3), and (-1, 3) on the graph to illustrate the horizontal line \(y = 3\).

Step 1: Understand the equation

The given equation is \(y = 3\). This is a horizontal line equation representing a constant function, where \(y\) is always 3 regardless of the value of \(x\).

Step 2: Substitute values for x

Choose three different values for \(x\) to find corresponding points on the graph. Let's select \(x = 0\), \(x = 1\), and \(x = -1\). Since \(y = 3\), each of these \(x\) values will have \(y = 3\). This gives us the points (0, 3), (1, 3), and (-1, 3).

Step 3: Plot the points

On a coordinate plane, plot the points (0, 3), (1, 3), and (-1, 3). All these points lie on the horizontal line \(y = 3\).

Step 4: Draw the graph

Using the plotted points, draw a straight horizontal line through them. This line is the graph of \(y = 3\) and continues infinitely in both the positive and negative x-directions.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface that helps us to visually represent equations. It is composed of two perpendicular axes, the horizontal x-axis and the vertical y-axis, which intersect at a point called the origin.

• The x-axis typically represents the independent variable, while the y-axis represents the dependent variable.
• Each point on the plane is represented by a pair of numbers known as coordinates, noted as \(x, y\).

These coordinates indicate the position of the point in relation to the origin.The coordinate plane is divided into four quadrants by the axes. The axes and the origin help in determining the position of points and guiding the drawing of lines or curves through those points. By plotting equations on this plane, we can see the graphical representation of data, trends, or mathematical functions. It's like a map where you can visualize concepts that may be abstract when just tackled with numbers.

Horizontal Line

A horizontal line is a straight line that runs from left to right across the coordinate plane. In mathematical terms, it means that no matter what x-value you have, the y-value remains the same.

• This occurs because, in a horizontal line, the slope is zero.
• The equation of a horizontal line is often given in the form \(y = b\), where \(b\) is a constant.

For example, in the equation \(y = 3\), the line runs horizontally through the y-axis at 3. A horizontal line is the graphical representation of a constant value for y.Horizontal lines are significant because they show the constant value of the dependent variable. They do not change as the independent variable varies.In real-world terms, this might represent a situation where something remains consistent over time, like a set interest rate or a flat fee.

Constant Function

A constant function is one of the simplest forms of function in mathematics. It is defined by an equation of the form \(y = c\), where \(c\) is a fixed number.

• No matter what value x takes, the output value \(y\) remains \(c\).
• Graphically, a constant function is represented as a horizontal line on the coordinate plane.

This type of function indicates that there is no relationship between the variables. An example of a constant function is the horizontal line \(y = 3\) from the original exercise. Here, the value of \(y\) will not change, even if \(x\) becomes a thousand times larger.Constant functions have practical applications in scenarios where something remains unchanging over time, such as a product's price stayed constant during a sale period, or as seen in equations reflecting enduring principles. Understanding constant functions helps in recognizing patterns and stability in functions, serving as a foundation for grasping more complex mathematical relationships.