Functions and Modeling

Functions and Modeling Assessment

Unit Homework Assignment 1

SHOW ALL WORK AND ANSWER ALL QUESTIONS ON SEPARATE PAPER. This is a non-calculator assignment.

1. Compare the following two velocity graphs. Which person went the farthest? Explain your details carefully. (Assume the scale is the same for both graphs)

2. Consider the graph below as a graph of position vs. time, and sketch the corresponding velocity graph. Then consider the original graph as a velocity graph and sketch the corresponding position graph. Lastly , consider the original graph as an acceleration graph and sketch the corresponding velocity graph. Which direction do you think is more difficult? Why? (put your answer on separate sheet of paper). When thinking “differentially” recall what happens at a cusp.

3. Give a “real world” example or application of a phenomena modeled by each of the following types of functions (this does not mean give an algebraic expression for each.). Explain why you chose the example and explicitly state the domain and range units used for your example (example: Domain – time measured in seconds; Range: height in meters).

a) linear b) quadratic c) exponential d) periodic

 2008 The University of Texas at Austin Functions and Modeling Homework Assignment 1

4. A particle starts from rest at the point two units from the origin and moves along a flat track with a constant positive acceleration for time t  0 . Which of the following could be the graph of the position s( t ) of the particle relative to the origin as a function of time t? Justify your choice.

5. In each region shown below, draw a section of a position (P) vs. time (T) graph that corresponds with the descriptions below. Assume moving to the right is away from CBR (motion detector) and moving to the left is towards CBR. Make sure to draw in the section of the graph that corresponds to the letter of the description.

a) The subject stood still. b) The subject walked faster and faster to the right. c) The subject walked to the right at a constant speed. d) The subject slowed down. e) The subject waked to the left faster and faster. f) The creature moved at a constant speed to the left. g) The creature stopped.

6. Without graphing, find the roots of x2 +4 x - 12 three ways. Show all work for each method.

of 5 Functions and Modeling Homework Assignment 1

7. Find the roots for and graph (including the complex plane) both branches of the quadratic f( x )= x2 - 3 x + 4 when considering a domain for the function that includes complex numbers and whose range consists of only real numbers.

8. Two cars are on a straight track. Car A starts at the beginning of the track at time 0 and moves forward. Car B starts at the beginning of the track one minute later and moves forward. The graph below shows the two cars’ velocity as a function of time. Consider only the pictured domain.

velocity Car B

time a) Based on the given domain only, which car travels farther? Why? b) Based on the given domain only, do the cars pass each other? Why or why not?

9. In your own words state what is meant by an injective function, surjective function, and bijective function. You may use examples to help illustrate the concepts as long as they are not examples already discussed in class.

10. Let f: X Y , be a function, and suppose that x0挝 X and y 0 Y satisfy

(x0 , y 0 ) f . If (x , y )喂 f , and x x0 , then y y 0 . State and justify whether you think this statement is true or false. If false, rewrite the sentence to make it a true statement.

11. Derive a formula for an equation of a circle with radius r and center (a,b).

12. For which of the following is it reasonable to model the relation as a function? Explain why using the definition and properties of a function. a. T is the temperature measured at time t. Is T a function of t? Is t a function of T?

b. D is the position of Tom’s car along Interstate I35 as measured by the mile marker sign posts and t is the time measured in seconds since midnight. Is D a function of t? Is t a function of D? c. Seven people, P, walk into a room with 28 chairs, C, and each sits in a chair. Is C a function of P? Is P a function of C? d. Twelve people, P, walk into a room with twelve chairs, C, and each takes a chair. Is C a function of P? Is P a function of C?

13. Give the definitions of the three conic sections.

14. Derive the “vertex form” formula for a parabola.

15. Derive the standard formula for an ellipse centered at the origin.

16. Sketch graphs of the relations and state what kind of curve you get. a. x2/25 + y2/9 = 1 b. x2/25 - y2/9 = 1 c. x2/25 + y2/9 = 0 d. x2/25 - y2/9 = 0 e. y = 3x2 +12x – 3

17. Write the equation of the ellipse with foci (-4,0) and (4,0) and where the constant in the definition of ellipse is 10.

18. Describe the symmetry of a parabola, ellipse, and a hyperbola. Explain why they have these symmetries in two ways. First use the definition and geometry and then use their equations and algebra.

19. Use geometry or algebra to show that all points on the ellipse x2/a2 + y2/b2 = 1 are in the rectangle with x bounded by –a and a and y bounded by –b and b.

20. Use geometry or algebra to show that no point on the hyperbola x2/a2 – y2/b2 = 1 is in the band with x between –a and a. What about the band where y is between –b and b?

21. Consider only the part of the hyperbola from the previous problem that is in the first quadrant. With this restriction, explain why we have a function. Now solve for y to find the formula for this function f(x). Show the limit as x approaches infinity of (f(x)- (b/a)x) is 0. Explain why this limit says the hyperbola has y = (b/a)x as an asymptote.

22. Find the center of each ellipse. Also determine the length of the major and minor axes. a. 4x2 + y2 -24x -4y +36 = 0 b. 9x2 +4y2 +18x -24y +44 =0 23. The set X has 5 elements and the set Y has 8 elements. a. How many functions are there from X to Y? b. How many injective functions are there from X to Y? c. How many surjective functions are there from X to Y?

24. A club with 23 members is about to elect a president, vice president, secretary and a treasurer. a. How can you model the election as a function? Identify the domain and range and how the function is defined. b. In terms of the election, describe what it means if your function in part a is injective. Describe what it means if your function is not injective. c. By counting functions, determine how many different outcomes the election could have assuming that any given person can hold at most one office. d. Now assume that a member is allowed to hold more than one office and count how many outcomes the election could have. 25. Explain how shuffling a deck of cards can be modeled as a function. Use the model to count the number of different orders a shuffled deck of cards could have.