Question

For each equation make a table of point pairs, taking integer values of \(x\) from -3 to 3, plot these points, and connect them with a smooth curve. $$y=-x+2$$

Short answer

The table of ordered pairs is: (-3, 5), (-2, 4), (-1, 3), (0, 2), (1, 1), (2, 0), (3, -1). These points are then plotted and connected with a straight line to graph the equation y = -x + 2.

Step-by-step solution

Create a Table of Values

Set up a table with two columns, one for x-values and one for y-values. Choose x-values from -3 to 3, since those are the integer values specified for the x variable. For each x-value, compute the corresponding y-value by using the equation y = -x + 2.

Calculate y-values

Substitute each chosen x-value into the equation to solve for y. Perform the calculation -x + 2 for each x-value to find the associated y-value. Fill these y-values into the table next to the corresponding x-values.

Plot the Points

Take the pairs of x and y from the table and plot them on the coordinate plane. Each pair (x, y) represents a point where x is the horizontal axis value and y is the vertical axis value.

Connect the Points

After all the points are plotted on the graph, draw a smooth curve that connects the points. Since the equation is a linear equation, the points will line up straight, and the curve will actually be a straight line.

Key concepts

Table of Values

Understanding the table of values is a fundamental step in graphing linear equations. It is a systematic way of organizing pairs of numbers, specifically input-output pairs that you can get from a given function or equation. Essentially, creating a table of values is about choosing a range of x-values and then using an equation to find the corresponding y-values.

In the context of the given exercise, where you have the equation \(y = -x + 2\), the process begins by selecting x-values within the range of -3 to 3. The equation dictates the relationship between x and y, so when you substitute the chosen x-values into the equation, it yields the y-values. These pairs of numbers are then listed in the table in two columns—one for x and another for y—which allows for an organized and visible representation of how the x-values relate to the y-values. This step is vital to simplify the process of plotting the points on a coordinate plane.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where points are defined by a pair of numerical coordinates. These coordinates signify the position of a point in terms of distance from two intersecting, perpendicular lines called axes. The horizontal line is known as the x-axis, and the vertical line is the y-axis. The point where they intersect is the origin, designated as \((0,0)\).

The plane is divided into four quadrants, each with a specific sign combination for the coordinates: Quadrant I (positive, positive), Quadrant II (negative, positive), Quadrant III (negative, negative), and Quadrant IV (positive, negative). Plotting a point requires you to move from the origin right or left along the x-axis and then up or down along the y-axis. For example, to plot \((3, -2)\), move three units to the right and two units down from the origin. Practicing this method enhances your ability to visualize and understand the spatial relationships of mathematical concepts.

Linear Equation Graphing

Graphing a linear equation involves translating an algebraic representation into a visual one. This visual is a straight line on the coordinate plane which showcases how the x and y values relate to each other according to the given equation. For the equation \(y = -x + 2\), after creating the table of values and then plotting the points derived from it, the graphing stage sees these individual points connected to display the equation's solutions graphically.

Every point on the line is a solution to the equation, meaning if you substitute the x and y from any point on the line into the equation, the equation will hold true. Linking the points should yield a straight line as it's a linear equation. The slope of the line, which in the given equation is -1, reflects how steeply the line inclines or declines. The y-intercept, here is 2, indicates where the line crosses the y-axis. Understanding these concepts not only helps in plotting linear equations accurately but also in interpreting the graphical data and how changes in an equation may affect the line's position on the coordinate plane.