Chapter 2: Functions and Models

Sarah D.

Discrete Mathematics

Quote on Courage:         "You gain strength, courage and confidence by every experience in which you really stop to look fear in the face. You must do the thing which you think you cannot do."   ~Eleanor Roosevelt~

    This picture relates to courage because high school is a very scary, and stressful time, filled with grades, projects, tests, and college applications. Through my busy schedule, I looked fear straight in the eyes, and made it through to my senior year, succeeding in my academics, music, and making plans for my future.. I stood up to the challenges I have faced, not only academically, but socially as well. Because of my courage through difficult times, I have built up valuable character traits that will be a huge asset to me this fall when I begin the hard journey through college, and later on in life when I start a family. I look forward to graduation, and I am proud of everything I have accomplished and overcome in the past four years. 

Section 2.6: Quadratic Models

Quadratic models are derived from quadratic models. The quadratic formula is :

-b + or - the square root of b squared minus 4ac, all over 2A. The standard set up for a quadratic function is      ax squared + bx + c

Example: If you have the function: f(x)= x squared - 3x - 1, find the y and x intercepts.

The y intercept is -1. To find the x intercept, you plug in 0 for y, and solve. for our problem we get                       0=x (squared) - 3x - 1

Your answer is plus/minus the square root of 5.

Section 2.7: Finding Quadratic Models

Given the following table, find the quadratic function. ax (squared) + bx + c

First, pick three equally spaced points. 

(1, 62), (4,92), and (7, 76)

sub into your formula, you end up with

62= a + b + c

92= 16a + 4b + c

76= 49a + 7b + c

when you solve this system of equations, a= -0.953, b= 15.875, and c=47.0833

Final step, plug that into your equation and you get: -0.9583x (squared) + 15.875x + 47.0833

Section 2.5 and 2.8: Correlation and The Men's Mile Record

Given the following table and graph, answer these questions.

Does this graph have a correlation? If so, is it strong or weak? positive or negative? 

Find the line of best fit.

Using the line of best fit, predict what Janie's time would be for the 15th meet.

   Janie's Swim Times in Seconds for the 50 Meter Butterfly Stroke

The graph has a strong negative correlation (because the graph is going down, and would all be very close to a line of best fit).

To find the line of best fit you use the formula f(x)=mx + b.    In order to find the line of best fit, you must pick two coordinates. 

(1, 42.7) and (10, 32.1) are the two coordinates I chose. You must make a system by plugging in the x and y coordinates in to the formula of y=mx + b.

42.7= 1m + b

32.1= 10m + b

to solve this, you subtract the functions, and get -9m=10.6. Therefore m= -1.178

Then you must plug the value of m into one of your equations (I'm plugging it into 42.7=1m + b because it will be easier to solve) to find the value of b. 

Work for finding b: -1.178 + b = 42.7

                           therefore b = 43.878

You take the values you found for b and a, and plug it into the equation of y=mx + b to find your formula for the line of best fit. In our case, our formula equals

y= -1.178x + 43.878

In order to predict Janie's swim time for the 15th meet, you simply plug 15 into the x spot of your equation (because in our graph, x is the meet number).

y= -1.178 (15) + 43.878

    y= 26.208

Therefore, if Janie continues her trend of swimming, by the time she reaches her 15th meet, her time for the 50 meter butterfly will be down to 26.208 seconds. 

The following website is helpful in dealing with this chapter on linear and quadratic equations: http://members.aol.com/hostaacpae/index.

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