It seems to me that students completing College Algebra should be able to answer the following sequence of questions. However, my students can’t and I bet your students can’t. Why not? Maybe we aren’t insistent enough during our courses.

Sketch all graphs on the same coordinate system. Label all important points with appropriate coordinates.

The ends of a diameter D of a circle C are the points (6, -1) and (-3, -5).

- What is the length of the diameter D of the circle C?
- What is the radius of the circle C?
- What is the center of the circle C?
- What is the equation for the circle C?
- Sketch the graph of the circle C.
- What are the x-intercepts of the graph of the circle C?
- What are the y-intercepts of the graph of the circle C?
- What is the slope of the line which contains the diameter D?
- What is the equation of the line L which contains the diameter D?
- What is the x-intercept of the line L?
- What is the y-intercept of the line L?
- What is the equation or inequality whose graph is the interior of the circle C?
- What is the equation or inequality whose graph is the exterior of the circle C?
- Explain why the point (3, -9) must be a solution to exactly one of:

i. the equation of the circle C,

ii. the inequality obtained by replacing = with < in the equation of the circle C, or

iii. the inequality obtained by replacing = with < in the equation of the circle C,

- What is the equation of the line S through the center of the circle C with slope 2/3.
- What is the x-intercept of the line S?
- What is the y-intercept of the line S?
- Sketch the graph of the line S.
- Suppose the center of the circle slides along the line S in an upward direction for a distance of ½ its radius. Call this new circle E.
- What are the coordinates of the center of the circle E?
- What is the equation of the circle E?
- Sketch the graph of the circle E?
- What system of equations or inequalities describes the region inside both circle C and circle E?
- What is the equation of the line K containing the centers of both circle C and circle E?
- What is the equation of the perpendicular bisector of the line segment joining the centers of circle C and circle E?

Granted this list is beginning to get a little ridiculous. Answering all of these questions would certainly be tedious and the numbers might be a mess (even with calculators).

The concepts involved are simple and some students know most of these concepts by the time they are completing College Algebra, but very few can use these concepts outside the arena of typical contrived one-concept textbook questions. Hardly any students have learned to use deductive reasoning but instead have learned to respond with certain “moves” when presented with stimuli of a certain form.