The Coordinate Plane, Graphs & Graphing Calculators

Tom Kelliher, MA 114

Sept. 11, 1998

Announcements: Be sure to bring books, calculators. Homework questions?

From last time:

1. Real numbers, solving quadratic equations.

Outline:

1. The coordinate plane: distance, midpoint formulas, graphs of equations.

2. Graphs & graphing calculators: foibles and follies.

Assignment: Read 1.5--6.

The Coordinate Plane

1. Cartesian (rectangular) coordinate system. Origin. X and Y axes. Quadrants.

2. The coordinates of a point are an ordered pair. Which order?

Distance Between Points

1. Consider the points and . What is the distance between them?

2. The distance, D between two points and is:

   

3. Why? (Recall Pythagoras' theorem.)

The Midpoint Formula

1. The midpoint of the line segment from to is:

   

2. Why? We must show that and . Why is that sufficient?

3. Examples: 1.3.5, 7, 17.

Graphs of Equations

1. A graph of a function in two variables is the set of points whose coordinates satisfy the equation.

2. Consider the circle with center at and radius r. Using the distance formula, derive its equation: .

3. What's the equation for the circle of radius 5 centered at the origin?

4. By completing the squares, find the center and radius of this circle:

   

5. Examples: 1.1.19, 23, 29.

Graphs & Graphing Calculators

1. Viewing windows: Seeing too much, seeing too little.

2. Try the exploration on Pg. 35.

3. Experiment: Set your calculator's window to Xmin = 0, Xmax = 3, Ymin = -4, Ymax = 0. Then, graph . Just looking at the graph, what type of a curve do you think it is? Set the zoom to Zstandard and check your guess.

4. Observations?

5. Aspect ratio --- making circles look like circles. Using Zstandard, graph the upper half of the circle centered at the origin of radius 4: . The circle isn't circular. Why? How do we fix it? Use Zsquare. Now, the curve doesn't touch the X-axis. Move the cursor to X = 1.9, Y = .1 and zoom in. Does the curve touch the axis now? Why not?

6. How does the calculator graph? Starting from window size, pixels in display, and equation.

7. Digitizing and aliasing --- Using Zstandard, graph . This function is defined for every value of x and only asymptotically approaches 0, but they all aren't graphed. Why?

8. Complete graphs: characterize the behavior of the equation. Examples: parabola, sine wave. Important to zoom in, out, and around to be sure you have the complete behavior.

9. Be sure to read Graphing Convention on pg. 45.

10. Examples: 1.4.18, 19.