Problem 45 - Algebra 1

Question

Use a table of values to graph the equation. \(y=-1\)

Step-by-Step Solution

Answer

The graph of \(y = -1\) is a straight horizontal line crossing the y-axis at -1. All the points on the line have the y-coordinate as -1.

Step 1: Understand the equation

The equation \(y = -1\) is a horizontal line where all the points on the line have the y-coordinate as -1.

Step 2: Construct a table of values

Choose several values for x to be able to draw the line. The values of x can be anything so it's common to choose simple numbers like -2, -1, 0, 1, 2. For each x, substitute x in to the equation to find y, here \(y = -1\) for all x-values.

Step 3: Plot the points

Plot points using the values from step 2 on a graph grid: (-2,-1), (-1,-1), (0,-1), (1,-1), (2,-1)

Step 4: Draw the graph

Finally, connect the points that have been plotted to create the graph for the equation

Key Concepts

Horizontal Line

A horizontal line in a graph is a straight line that goes from left to right or right to left in a consistent manner across the graph. The defining characteristic of a horizontal line is that it has the same y-coordinate for all its points. This means that no matter where you are on this line, its height or position vertically does not change.

In mathematical terms, the equation of a horizontal line can be expressed as:

• Constant y value, like in the equation \( y = -1 \)
• Line is parallel to the x-axis
• No slope, or the slope is zero

Understanding horizontal lines is important, because recognizing their constant nature can help simplify many graphing tasks. Once you establish the y-value, such as -1 in our example, you can easily plot it by drawing a straight line through points with the same y-coordinate. This is especially useful in identifying patterns or solving equations that require graph interpretation.

Coordinates

Coordinates are essentially a set of values that show an exact position on a graph. In a two-dimensional space, they are given as pairs of numbers. The first number is the x-coordinate, representing the position along the horizontal axis, and the second is the y-coordinate, pointing out the position along the vertical axis.

To graph an equation like \( y = -1 \), you use these pairs to find precise points on the graph:

• For the equation \( y = -1 \), every pair has a y-coordinate of -1.
• Choosing x-coordinates, such as -2, -1, 0, 1, 2, helps to create points like (-2, -1), (0, -1), and (2, -1).
• You can plot these points on the graph to form a horizontal line.

Using coordinates is essential in graphing as it provides clarity and precision in positioning points. Hence, understanding how to work with coordinate pairs is crucial for accurately graphing any equation.

Table of Values

A table of values is a useful tool when graphing equations. It is a structured way to organize the x and y pairs you want to plot on your graph. To construct a table of values, you generally follow these steps:

• Select a range of x-values. For instance, you might choose values such as -2, -1, 0, 1, and 2 for simplicity.
• Apply the equation to each x-value. In the equation \( y = -1 \), for every x-value, the corresponding y-value will always be -1.
• Write down the coordinate pairs, like (-2, -1), (-1, -1), and so on, in a list or table format.

The benefit of having a table of values is that it helps keep your work organized and easy to understand. It provides a quick reference to ensure that plotting is accurate, and it simplifies the process of graphing lines and curves by offering clearly defined points to connect.