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Jul 25­12:43 PM

1.1 Modeling and Equation Solving

Mathematical model: mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior.

We'll concentrate on three types of mathematical models:1.)2.)3.)

Now, it's time for a scavenger hunt! Once you find your matches, write down your information below.

Numerical:

Algebraic: Graphical:

Jul 30­3:23 PM

Year Long Jump High Jump

0 282.875 74.8 4 289 71 8 294.5 75 12 299.25 76 20 281.5 76.25 24 293.125 78 28 304.75 76.375 32 300.75 77.625 36 317.3125 79.9375 48 308 78 52 298 80.32 56 308.25 83.25 60 319.75 85 64 317.75 85.75 68 350.5 88.25 72 324.5 87.75 76 328.5 88.5 80 336.25 92.75 84 336.25 92.5 88 343.25 93.5 92 342.5 92

Olympic Gold Medal Performances

The modern Olympic Games are a modified revival of the Greek Olympian Games that came to be largely through the efforts of the French sportsman and educatorBaron Pierre de Coubertin. The Games are an international athletic competition that has been held at a different siteevery four years since their inauguration in 1896, with occasional interruptions in the times of world wars.

The data for the gold medal performances in long jump andhigh jump are given to the right. Distances are measured ininches. Year 0 represents 1900.

1.) According to the numeric model, what has been thetrend in the long jump over the years?

2.) In what 4 year interval did the gold medal performancefor the high jump change the most?

3.) Let's make some scatter plots.

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Jul 30­7:21 PM

4.) Does the data seem to follow a linear model?

Long Jump: Use (4,289) and (88,343.25) as points to find an equation of the line and superimpose it onto the scatter plot.

High Jump: Use (20,76.25) and (92,92) as points to find an equation of the line and superimpose it onto the scatter plot.

5.) If the jumps continue to follow the linear models, find the gold medal long and high jumps (in inches) in the 2012 Olympics.

Jul 30­7:44 PM

Comparing Pizzas: A pizzeria sells a rectangular 20" by 22" pizza for the same price as its large round pizza that's 24" in diameter. If both pizzas are the same thickness, which option gives the most pizza for the money?

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Jul 30­7:50 PM

Elapsed time (sec) 0 1 2 3 4 5 6 7 8Distance traveled (inches) 0 0.75 3 6.75 12 18.75 27 36.75 48

What graphical model fits the data?

Can you find an algebraic model that fits?

Galileo Galilei (1564-1642) spent a good deal of time rolling balls down inclined planes carefully recording the distance they traveled as a function of elapsed time. His experiments are commonly repeated in physics classes today.

Jul 30­8:31 PM

Factoring Review: Factor the following completely over the real numbers.

1.) x2 - 16 6.) x2 + 2xh + h2

2.) x2 + 10x + 25 7.) x2 + 3x - 4

3.) 81y2 - 4 8.) x2 - 3x + 4

4.) 3x3 - 15x2 + 18x 9.) 2x2 - 11x + 5

5.) 16h4 - 81 10.) x4 + x2 - 20

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Jul 30­8:15 PM

If a is a real number that solves the equation f(x) = 0, then these three statements are equivalent:

1.) The number a is a _____ (or ________) of the equation f(x) = 0.

2.) The number a is a _____ of f(x).

3.) The number a is an __________ of the ______ of y = f(x).(Sometimes the point (a,0) is referred to as an x-intercept.)

Solving Algebraically

1.) x(2x - 1) = 10

Jul 30­8:35 PM

2.) (x+11)2 + 5 = 126

3.) x2 - 7x - (3/4) = 0

4.) √x + x = 1

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Jul 30­8:35 PM

Solving Graphically Solve the equation graphically by converting it to an equivalent equation with 0 on the RHS and then finding the x-intercepts.

1.) x + 1 = x3 - 2x - 5

2.) x2 = x

3.) 3x - 2 = 2 √x + 8

Jul 30­8:38 PM

1.) Prove that 2 is a factor of n2 + n for every positive integer n.

2.) Prove that 6 is a factor of n3 - n for every positive integer n.