My Question is... On our exercies you assigned today, Aug. 31... I do not understand the
 directions... Do we use the same formula we used for the water depth expirment? And do we
 only do Data set 1 and 2 or should we read ahead of that?
 
 Good question. This one was answered in an email sent to everyone in the class.
 
 
 Hey dave, I'm not confused about anything but I would like to know when solving for c after
 you have unknowns a and b does it matter which equation you solve it in?
 
 If you've solved correctly for a and b, then you'll get the same thing no matter which
 equation you use.
 
 Similarly if you've eliminated c, then eliminated b from the two resulting equations, and
 then finally solved for a, it doesn't matter which of the two equations you substitute a
 into, you'll get the same result for b.
 
 Once you've used your values of a and b to get c from one of the original equations, you
 should then substitute your a, b and c into another one of the original equations to check
 your work.
 
 Another way of checking, of course, is to get your model y = a t^2 + b t + c and then to
 substitue your chosen t values. Each of your chosen t values should give you the
 corresponding y value.
 
 
 
 What exactly do you mean when you say the word model, because you sometimes refer to a graph
 or a mathimatical equation, and when you say model it doesn't specify which one you want, a
 graph or an equation?
 
 The model is the function, which is represented by both the graph and the model.
 
 The function can usually be expressed as a graph or an equation.
 
 For example the depth vs. clock time function might be y = f(t) = .41 t^2 - 3.74 t + 94.
 The equation is y = .41 t^2 - 3.74 t + 94, and the graph represents this function.
 
 The graph represents the equation, and the equation represents the graph, so the two are
 actually identical. The equation is generally more accurate than the graph, but within
 limits of accuracy the equation and the graph tell you exactly the same thing. What they
 both represent is the function.
 
 
 
 If T=.23L^.48 and that is a constant then can those numbers always be put into any equation,
 like T=A*L^p?
 
 y = A x^p is the general form of a power function y of the independent variable x.
 
 Our pendulum period vs. length function uses variables T and L instead of y and x.
 
 So T = A * L^p is the general form of a power function T of the independent variable L.
 
 Knowing that the pendulum period should be a power function of its length, we would then
 know to use the form T = A * L^p. If we know the values of T at two different values of L,
 we can then substitute each pair of values into the form to get an equation in the
 parameters A and p (see notes from late in the first week for specific examples of this).
 
 Once we have two equations in A and p, we can solve them for A and p.
 
 Having obtained our values of A and p, we then plug them into the form T = A * L^p, giving
 us our mathematical model of period T as a function of length L.
 
 The model we got from our observations in class (again see notes from week 1) is pretty
 close to T = .23 L^.48.
 
 The ideal model given by physics is T = .20 L^.50. So we did pretty well.
 
 
 
 
 My question is... When you say that we need to memorize the above outline of the model, are
 you talking about model part C(Validate and use the model)? or model part A(Obtain and
 represent Date), B (Obtain a Model), & C(Validate and use the model)?
 
 This is the outline you are expected to memorize, and which will be used throughout the course.
 
 Summary of Modeling Process, Version 3
 A. Obtain and Represent Data
 
 A1. Orient
 
 A2. Observe
 
 A3. Organize Data
 
 A4. Graph
 
 
 
 B. Obtain a Model
 
 B1. Postulate
 
 B2. Select Representative Points
 
 B3. Obtain an equation for each selected point
 
 B4. Solve the system of equations
 
 B5. Substitute parameters
 
 C. Validate and Use the Model
 
 C1. Graph the model
 
 C2. Quantify the comparison
 
 C3. Pose and answer questions
 
 C4. Do the science: relate the mathematics to the real world.
 
 To Be Memorized
 The above outline of the model is to be memorized. At any time during the course, on any test or quiz, you might be asked to write down this model and illustrate the meaning of one or more of the steps.
 
 This outline will be more or less followed throughout the course.
 
 
 What is the vertex exactly and how exactly do we find it?
 
We will see more about this in the last of the worksheets in Wednesday's handout, the Assignment 3 worksheet "Properties of Quadratic Functions".
 
 A brief answer is:
 
 y = a t^2 + b t + c is the equation of a parabola. The graph of this equation has the shape of a parabola (a parabola is a curve with the property that every point of the curve lies at the same distance from its focus and directrix, but the details of that definition and how it is related to the equation given here is a second-semester topic).
 
 The parabola is symmetric about a vertical line. The point where the parabola intersects this vertical line is called the vertex. It is either the high or the low point of the parabola, depending on whether the parabola opens upward or downward.
 
 The axis of symmetry is the line t = - b / ( 2 a), where a and b are the coefficients of t^2 and t.
 
 You can find the y coordinate of the vertex by substituting this t value into the equation of the parabola.

basic precalculus functions with parameters.xls

Graph 3a

Graph 4a

Graph 1b

Graph 2b

Graph 3b

Graph 4b

Graph 5b

Graph 6b

Graph 7b

Graph 8b

Graph 9b