At the end of this lesson, students will be able to:
- Identify coordinates of points.
- Plot points in a coordinate plane.
- Graph a function given a table.
- Graph a function given a rule.
Terms introduced in this lesson:
x− and y−axes
x and y coordinates
positive x, negative x
positive y, negative y
graph of a function
Teaching Strategies and Tips
Introduction: Motivate xy−coordinates with examples from daily life that employ rectangular coordinate systems.
- Examples: a city map, the game of Battleship, a chessboard, spreadsheets, assigned seating at a theater.
- Discuss how to find a particular location in each example: a seat in a theater can be found by row number and then by seat number.
- Point out that the examples are lattices, differing from the Cartesian coordinate system in that they are discrete.
Use Examples 1-3 to demonstrate finding coordinates of points on a graph and Examples 4 and 5 to plot points given their coordinates. Allow the class to make observations such as:
- The coordinates of a point cannot be interchanged since the first coordinate specifies going left/right and the second coordinate, up/down. For example, (2,7) is not the same point as (7,2).
- If a coordinate of a point is 0, then the point resides on an axis.
- Quadrants can be distinguished by the signs of the coordinates contained in them. For example, a point having coordinates with the signs (−,+) resides in quadrant II. Points with coordinates having signs (−,−) belong to quadrant III.
- In Example 4, it is necessary to display four quadrants so that all points will be visible. The set of points in Example 5 have only positive coordinates; it is convenient therefore to display only the first quadrant. As an informal rule, axes do not need to be extended farther than the largest and smallest x−coordinates and y−coordinates.
- Resize a graph by rescaling the axes. In general, the x and y−axes can be scaled differently. Axis tick marks do not need to be unit increments.
General graphing tips:
- In applied problems, the independent and dependent variables should be distinguished early. Ask:
What quantity is depending on the other?
- In setting up the axes, a suitable scale must be chosen. Ask:
Will the important features of the graph be visible?
Will it be necessary to use different increments along the two axes?
- Constructing tables is a valuable tool. See Examples 6 & 7. Allow students to use the simple inputs, x=0,1,−1,2, in their tables when appropriate.
The second method in Example 7 will be returned to in greater detail in a subsequent chapter.
General Tip: To determine the graph of a linear relationship, no more than two points are needed. Students can be encouraged to plot at least three to ensure no arithmetical errors were made.