PRACTICE FINAL - 1010-004, FALL 2013 If you are completing this practice final for bonus points, please use separate sheets of paper to do your work and circle your answers. Turn in all work you did to arrive at the answers. Answer all numbered questions in this practice final to receive the full bonus points. You can get partial points by answering some questions. What will NOT be on the final The following is a list of concepts from chapters 1-9 in the book which we either did not cover in class, or which are not important enough that you should worry about them for the final exam. This is not a complete list of things that will not show up, but is a good start. This is for your information only, and there is nothing you need to do here to get your bonus points for the practice final. Assume that everything from Chapters 1-9 EXCEPT what is written here is fair game for the final. - Section 1.3 - Names of the basic Algebraic properties (for example, Commutative Property of Multiplication). You still need to know how to apply the properties, though. - Section 2.3 - Business and Scientific Problems. The only thing you need from this section is Distance = Rate x Time. - Section 2.3 - Area / Volume Formulas. You should know the perimeter and area of a rectangle or 3-d box. Any other needed formulas will be provided. - Section 2.5 - Absolute value equations and inequalities will not be on the final. - Section 3.1 - You do not need the Midpoint formula. - Section 3.5 - Not covered in class. - Chapter 4 - You may always solve any system using either substitution or elimination, whichever is more convenient for you. You will not have to solve a system graphically. - Section 4.1 - We did not talk about the terminology consistent, dependent, and inconsistent to refer to systems of equations. - Section 4.1 - We did not talk about the business applications. - Section 4.3 - You will not see a system with more than 2 variables. - Section 4.4 - Not covered. - Section 4.5 - Not covered. - Section 4.6 - Not covered. - Section 5.1 - We did not discuss scientific notation in detail. - Section 5.4 - You will not need to memorize the factoring formula for Sum or Difference of Cubes (x 3 a 3 or x 3 + a 3 ) - Section 6.4 - Not covered. - Section 6.5 - It is best to just know how to do long division of polynomials and not have to memorize synthetic division. Synthetic is really not necessary if you know long division. - Section 6.7 - Not covered. 1
- Section 7.3, 7.4 - You will only need to deal with radicals when solving radical equations or quadratic equations, so these two sections will probably be irrelevant. - Section 7.6 - Not covered. - Chapter 8 - Many of the book examples and homework problems in this chapter have complex solutions (imaginary numbers). Ignore these since we did not cover section 7.6. - Section 8.4 - Not covered. Note that you are still expected to know the basic shape of quadratic equations (a parabola) and should be able to sketch a graph using a table (Section 3.2). You will not be asked to find the vertex of a parabola. - Section 8.5 - Concentrate only on the problem types I gave in homework for this section. - Chapter 9 - We did not define the term asymptote in class. We did discuss how the function f(x) = a x approaches the x-axis on the left, and how the function g(x) = log(x) approaches the y-axis. - Chapter 9 - We did not explicitly discuss any applications in class, but you may be expected to solve an application equation with exponents or logs. - Section 9.3 - We did not cover the Change of Base formula. This formula is only useful for using your calculator to calculate logs with base other than 10 or e. Any logs you may have to calculate on the final will be base 10 or e. - Section 9.6 - Not covered, but you might expect to see problems like these on the final. After plugging in numbers they are exactly the same as problems from section 9.5. - Chapters 10, 11 - Not covered. Concepts The final exam has some questions that are meant to test how well you understand the concepts in this course. Some of them may just be short answer questions, and others may be problems you know how to do, but phrased a little differently than usual, or in a story problem instead of an algebraic problem. Here are some examples of what you might see. (This list is not meant to be exhaustive, but merely an example of the sort of questions you should know how to answer!) 1. How can you tell whether three points in the plane are on the same straight line? Explain why your method works. 2. How can you use the slope of a straight line to help you graph it? Also, how can you tell what the slope of a straight line is from its graph? 3. How is the solution to a system of equations represented graphically? 2
4. What is happening graphically if a system of equations does not have a solution? 5. Give an example with numbers to demonstrate that x + y x + y. 6. Give an example with numbers to demonstrate that log(x + y) log(x) + log(y) 7. Explain why someone should want to find the domain of a function. What does the domain represent? 8. In the simplification x2 = x, the domain restriction x 0 is necessary. Explain what x this domain restriction accomplishes and why it is necessary. 9. Along the same lines, explain why no domain restriction is necessary when simplifying x x 2 = 1 x. 10. What transformation is represented by going from f(x) = x 2 to g(x) = (x 1) 2? Explain how you know. 11. Describe the process for completing the square in a quadratic expression, x 2 + bx + c. 12. How can you tell from a graph whether the graph represents a function? How can you tell whether that function has an inverse? 13. How can you tell from a set of ordered pairs whether they represent a function? How can you tell whether that function has an inverse? 14. Explain why f(x) = x 2 and g(x) = x are not inverses. 15. How does the exponential function f(x) = 2 x behave when x is very large? When x is very large and negative? 16. How does the logarithmic function g(x) = log(x) behave when x is very large? When x is very close to 0? When x is negative? 17. How would you find the x-intercepts of a function? The y-intercepts? 18. Is it possible for a function to have more than one y-intercept? Is it possible for a function to have no y-intercept? 3
Problems 1. Solve: 6x + 10 = 4(x 7) 2. You are 6 feet tall and your shadow is 15 feet long. If a tree nearby has a shadow that is 40 feet long, how tall is the tree? 3. On a long trip you drive 400 miles at 60 miles per hour, and 300 miles at 70 miles per hour. What was your average speed for the trip? (Your average speed is the total distance divided by total time) 4. Solve: 4x + 9 19 5. Find the distance between the points ( 1, 4) and (5, 2). Simplify your answer. 6. Determine whether the points ( 4, 3), ( 1, 2) and (3, 9) lie on a straight line together. (NOTE: In section 3.1 we did this by finding distances between the points. You can also accomplish this by finding SLOPES between each pair of points and seeing if the slopes are all equal.) 7. Sketch a graph of the function f(x) = 2 x + x by making a table of values, plotting the points and drawing a smooth curve. 8. Find the x and y intercepts of the graph of g(x) = x 2 x 2. 9. Find an equation for the straight line through the points (1, 4) and ( 1, 7). Write your equation in slope-intercept form. 10. An antique table is worth $1400 after 25 years and $2000 after 40 years. If the relationship between price and time is linear, how much should the table be worth after 60 years? How much was the table worth initially (at 0 years)? 11. The equation 3x + 5y = 15 represents a straight line in the plane. Find its slope, x-intercept, y-intercept, and graph it. 12. Determine whether the points (1, 4), (5, 2), (7, 4), (3, 3), ( 1, 7) define a function. Explain your reasoning. 13. For f(x) = x 1, find the domain of f and evaluate f(5). x+3 14. For g(x) = 16 x 2, find the domain of g and evaluate g( 2). 15. Graph the function f(x) = 2 x. Then describe how the graphs of g(x) = 2 x+3 and h(x) = 2 x are different. Last, write an equation for a function t(x) which has the same graph as f(x) but shifted down by 3 units. 16. Solve the system of equations, 2x + y = 14, 3x 2y = 14 4
17. At an amusement park you can buy 5 child tickets and 3 adult tickets for a total of $25. You can also buy 9 child tickets and 4 child tickets for a total of $38. What is the cost of each ticket individually? 18. Simplify the expression, x 2 y 3 x 5 y 2 19. Simplify the expression, (xy 1 ) 2 x 5 20. Multiply the polynomials, (2x + 5)(x 2 x 6) 21. Solve the equation by factoring, x 2 13x + 36 = 0 22. Solve the equation by factoring, 3x 2 2x 8 = 0 23. Multiply the rational expressions and simplify, x2 1 x2 +6x x x+1 1 24. Add the rational expressions and simplify, + 2x 1 x+3 x 5 25. Divide the polynomials, x2 +7x+15 x+4 26. Solve the equation, 24 x 3x = 14 27. Solve the equation, 2x + 12 = x 28. Complete the square for the expression x 2 + 12x + 15 29. Solve the equation, x 2 + 5x + 2 = 0 30. A rocket launched from the top of a building has height function h(t) = 16t 2 +92t+96, where h(t) is measured in feet and t is in seconds. How high is the rocket after 2 seconds? When does the rocket hit the ground? What is the height of the building the rocket is launched from? 31. Solve the inequality, x 2 + 6x + 8 0 32. Solve the inequality, x+3 x 5 2 33. Define f(x) by the ordered pairs (3, 1), (2, 2), ( 1, 6) and define g(x) by the ordered pairs (2, 5), (4, 1), ( 1, 3). Then find (f g)( 1) and (g f)(2). 34. Define f(x) = 3x 2 and g(x) = x 2 3. Find (f g)(x) and (g f)(3). 35. Find the expression for the inverse of f(x) = 4x + 7. 36. Find the expression for the inverse of h(x) = x + 7 (Hint: You will need to restrict the domain of h 1 (x) to make it a true inverse) 37. If f(x) is described by the points (1, 5), (4, 3), (6, 7), then describe the function f 1 (x) if it exists. 5
38. Determine algebraically whether the functions f(x) = 3 x + 3 and g(x) = x 3 3 are inverses. 39. Solve the equation, 6 2 x+3 = 12 40. A sample of the radioactive element Cesium-137 begins to decay so that the amount of the material follows the function f(t) = 100 2 t/30, where f is in grams and t is in years. How much Cesium-137 do you begin with? How much do you have after 50 years? How long will it be before you only have 10 grams of Cesium-137 left? 41. Solve the equation, log x + log(x 2) = 3 6