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Magazine Chapter 2: Problem 37
Make a table of values and graph six sets of ordered integer pairs for each equation. Describe the graph. $$y=x+2$$
Short Answer
Expert verified
The graph of \( y = x + 2 \) is a straight line with a slope of 1, crossing the y-axis at (0,2).
Step by step solution
01
Choose x-values
Select six different integer values for the variable \( x \). A good choice can be values ranging from -3 to 2, which will provide a balanced distribution across the graph.
02
Calculate y-values
Use the equation \( y = x + 2 \) to find the corresponding \( y \)-values for each chosen \( x \)-value. For instance, if \( x = -3 \), then \( y = -3 + 2 = -1 \). Continue this for every selected \( x \)-value.
03
Create the Table
Record the \( x \)-values and their corresponding \( y \)-values in a table format. This table should include pairs such as \((-3,-1), (-2,0), (-1,1), (0,2), (1,3), (2,4)\).
04
Plot the Ordered Pairs
Using a coordinate plane, plot each ordered pair from the table. Begin with the point \((-3,-1)\) and proceed with all other points until \((2,4)\) is plotted.
05
Draw the Graph
Connect the plotted points with a straight line. Since the relationship between \( x \) and \( y \) is linear as indicated by the equation \( y = x + 2 \), the graph will be a straight line.
06
Describe the Graph
The graph is a straight line with a slope of 1 that crosses the y-axis at the point \((0,2)\). This indicates that for each unit increase in \( x \), \( y \) increases by 1. The line extends infinitely in both directions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Plane
A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It is made up of two axes: the horizontal axis, which is typically the x-axis, and the vertical axis, known as the y-axis.
These axes intersect at a point called the origin, which has coordinates (0, 0). By using a coordinate plane, we can visually represent mathematical concepts and relationships.
These axes intersect at a point called the origin, which has coordinates (0, 0). By using a coordinate plane, we can visually represent mathematical concepts and relationships.
- Each point on this plane is described as an ordered pair of numbers, (x, y).
- The x-coordinate refers to horizontal positioning, determining how far left or right a point is from the origin.
- The y-coordinate specifies vertical positioning, showing how far up or down a point is from the origin.
Table of Values
A table of values is a simple yet powerful tool for organizing and calculating numerical data points in algebra. When dealing with linear equations, such as
y = x + 2, a table of values helps identify specific points that fit on the equation's graph.
- First, choose several x-values; standard practice often starts with small numbers, including negatives, zero, and positives.
- Use the chosen equation to calculate the corresponding y-values.
- Fill out the table with paired x and y numbers, such as (-3, -1), (-2, 0), and so on.
Ordered Pairs
Ordered pairs are a fundamental concept in mathematics used to locate points in a coordinate system. Each ordered pair, given as
(x, y), uniquely identifies a point's position on the coordinate plane.
Ordered pairs function like addresses, indicating exactly where each point is located:
Ordered pairs function like addresses, indicating exactly where each point is located:
- The first number ( x) tells us the position along the horizontal axis.
- The second number ( y) indicates the vertical position.
Graphing
Graphing is the illustrative method of representing equations and mathematical relationships. It involves plotting ordered pairs from a table of values onto a coordinate plane.
After plotting points like (-3, -1), (-2, 0), (0, 2), we connect them to form a line, illustrating a linear relationship.
There are several steps to effective graphing:
After plotting points like (-3, -1), (-2, 0), (0, 2), we connect them to form a line, illustrating a linear relationship.
There are several steps to effective graphing:
- Begin by marking each ordered pair as a point on the plane.
- Make sure to accurately disperse all points according to the ordered pairs.
- Draw a straight line through these points to depict the equation visually.
- This line allows us to understand how y-values change with x-values.
Slope
Slope is a critical concept in understanding linear equations and their graphs. It measures the steepness or incline of a line and is represented by the letter m.
For the equation y = x + 2, the slope is 1, which simply means that for every unit increase along the x-axis, the y-value increases by the same amount.
For the equation y = x + 2, the slope is 1, which simply means that for every unit increase along the x-axis, the y-value increases by the same amount.
- A slope can be calculated from two points ((x_1, y_1) and (x_2, y_2)) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
- A positive slope means the line rises from left to right, while a negative slope descends.
- If the slope is zero, the line is horizontal, and if undefined, the line is vertical.
