Algebra II Study Guide Answer Key

ALGEBRA II – STUDY GUIDE 1.1-1.8 1. Imagine you are watching a racer running around a quarter mile oval track from the starting line at a constant rate. a. Name at least 3 quantities that vary. St Time, total distance traveled, distance from start, etc. b. Name 1 quantity that does not vary. Length of the track, runner’s speed (stated as constant) c. Construct a graph of the direct distance of the racer from the starting line in terms of distance the runner has traveled around the track. 2. Karl rides a ferris wheel. He boards at the bottom and rides a few times before getting off. The following graph represents his height above the ground in feet with respect to his total distance traveled around the ferris wheel. ∆y = 29 (approx.) a. The point (20, 15) is plotted on the graph. What does this point convey about the situation? ∆x = 40 After traveling 20 feet around the wheel, Karl is 15 feet off the ground. He has just started his ride and is less than half the way up to the top (almost half the way) b. How much does Karl’s distance around the ferris wheel change from 20 to 60 feet? 40 feet c. Represent the correspoinding change in Karl’s distance from the ground from 20 to 60 feet traveled. Appears to be approximately 29 feet. d. Use the graph to estimate his distance from the ground at 60 feet. After traveling a total of 60 feet, he will be approximately 44 feet off the ground. 3. Susan had a 93.1% in her history class at the beginning of July. Later that month, after taking a test, her grade was a 89.6%. What was the change in Susan’s grade? -3.5 4. Suppose we start with x=-15. What expression represents the change in x from x=-15 to any possible value of x? ∆x = x+15 5. Suppose Lauren bought a puppy. At the time she bought the puppy it weighed 9 pounds. Let w represent the weight of the puppy in pounds. a. What is the difference in saying w=5 and ∆w=5? W= 5 means the puppy weighed 5 pounds. ∆w =5 means the puppy’s weight change by +5 pounds (he gained 5 pounds) b. If we substitute w = 24.5 into the equation w-8, what are we calculating? 24.5 – 8 This is the change in weight from 8 pounds to 24.5 pounds 6. What does it mean to move at a constant speed? For an object traveling at a constant speed, the same change int ime elapsed will always correspond to equal changes in distance traveled. Likewise, the same change in distance traveled will correspond to equal changes in time elapsed. Also, if we know how far the object travels in some total amount of time, then in some fraction fo that time it will travel the same fraction of the total distance. 7. Paul is walking in the park. Assume that he walks at a constant speed during the entire trip. It takes him 5 seconds to walk the 20 feet between the tree and the waterfountain. a. How far can Paul travel in 1 minute (60 seconds)? 240 feet b. How long does it take him to walk 10 feet? 2.5 seconds c. What is his unit rate (feet per second)? 4 feet per second c. How far can Paul travel in 14 seconds? 56 feet 8. Marcy loves purchasing candy from the local store for $4 per pound. Let c represent the total cost of purchasing the candy (in dollars) and let w represent the weight of the candy purchased (in pounds). a. Suppose Mary has some trail mix in a bag already. She adds two more pounds of candy to the bag. Represent this change in the number of pounds of candy with ∆ notation. ∆w= 2 b. How much does the total cost of the candy purchsed change? (represent answer in ∆ notation). ∆c= $8 c. How much does the total cost of the candy purchased change if she adds 5 pounds? ∆c= $20 d. How much does the total cost of the candy purchased change if she removes 3.5 pounds? ∆c= $-14 e. If her friend hands her $25, how mandy additional pounds of candy can she add to the bag? ∆w = 6.25 pounds #9-11. Complete steps a-d for each problem below: #9) #10) X -6 -2 1 4 #11) (3,4) (-6, -8) Y 16 4 -5 -14 a. Find the constant rate of change b. Write a formula to describe the graph (in point-slope form) c. Convert the formula from the point-slope form into the slope-intercept form. d. Find the point on the graph where x=3. #9) a. m=-3 b. just one of the following: y =-3(x+6)+16 y=3(x+2)+4 y = 3(x-1)-5 y = 3(x-4)-14 #10) a. m=1 b. just one of the following: y =(x+4) - 3 y = (x-1)+2 #11) a. m=4/3 b. just one of the following: y =4/3 (x-3)+4 y = 4/3 (x+6)-8 c. y = -3x-2 c. y = x +1 c. y = 4/3 x d. (3,-11) d. d. (3,4) (3,4) 12. Suppose we know that y changes at a constant rate of 3 with respect to x. a. Write the above statement using ∆ notation. ∆y=3∆x If we start off at x=4 and let x change to be x=13, b. What is the change in x? ∆x=9 c. By how much does y change for the change in x you found in part (b)? ∆y =27 d. Suppose we know that y=5 when x= 4. What is the value of y when x = 13? y = 32 e. Plot the two points and show the changes in x and y on the graph to the right. See graph – the two points are (13, 32) and (4, 5). Then, show changes in x and y. 13. Given that ∆y = 2.4∆x, complete the following table of values: X -3 0 5 8 Y -10.2 -3 9 16.2 14. Plot the point (2,1) on the graph. Given that the ∆y = 1.5 ∆x a. What is the value of y when x=3.5? Represent your reasoning on the graph. 3.25, so there is a point at (3.5, 3.25) b. What is the value of y when x= -2. Represent your reasoning on the graph. -5, so there is a point at (-2, -5) 15. Given (2,1) and (8, 5) are points on the graph. What is the value of y when x=5? y =3. So, there is a point (5,3) on the graph. 16. Determine which table below is linear, then write the equation of the line: X Y -4 8 -1 6.5 4 4 6 3 Linear. y =-1/2 (x+4) -8 X -3 0 8 12 Not linear. Y -13 -4 20 28 17. Given that ∆y = -7 ∆x and that when x=4, y= 21. We want to know the new value of y for any value of x. Answer the questions that follow. y = -7 (x-4) + 21 a. What does x-4 represent? Change in x from x=4 to new x value b. What does -7(x-4) represent? Change in y from y=21 to new y value c. What does -7(x-4) +21 represent The y value for new value of x d. What does 9-4 represent? Change in x from x=4 to x=9 e. What does -7(5) represent? Change in y from y=21 to new point f. What does -35 + 21 represent? ∆y + y1, which is the new y value when x=9 18. Consider the formula y = 4x+2. Suppose we want to find the value of y when x=2.5. Explain how the formula determines the value of y using the meaning of constant rate of change. y = 4 (2.5) + 2 - The change in x from x=0 to x=2.5 y = 4 (2.5) + 2 - The change in y from x=2 to new y value y = 4(2.5) + 2 - The y value at y = 2.5 19. The constant rate of change of y with respect to x is -3.5 and (-3, 2) is a point on the graph. a. Write a formula for the linear function. y = -3.5 (x+3) +2 b. Find the value of y when x= 5. What is the new point on the graph? (5, -26) c. Convert the function in part a from the point-slope form into the slope-intercept form. y = -3.5x – 8.5 20. Given that y=3x-2, fill in the table with points on the graph. Come up with another point (of your own) that is on the graph. X Y -3 -11 0 -2 2 4 10 28 Answers for other point will vary. Given the the information below, 1) write the equation of the line in slope-intercept form, 2) finish the graph, and 3) identify the vertical intercept. 21. ∆y = -3/2 ∆x 1) y = -3/2 x – 4 22. ∆y = 5 ∆x 3) (0, -4) 1) y = 5x + 3 3) (0,3) 23. (4,5) (0, 3) y=½x+3 The vertical intercept is at (0, 3) Graph each line and label 3 points on the line (make sure one of the points is the y-intercept): 24. y = 0.75 (x-4)+2 25. y = x -5 25. What is the vertical intercept of y= 4(x-5)+2? (0, -18) 26. Write the equation and state the slope of each line: X = -4, m = undefined y=2, m=0 27. Micah rents out his bouncy castles for parties. The table below summarizes the cost of renting the bouncy castle for different time periods. Hours rented 2 6 12 24 a. Total cost of renting 260 380 560 920 There is a constant rate of change fo the total amount she charges (in dollars) with respect to the number of hours the bouncy castle is rented. Determine the constant rate and explain what it represents in this context. Constant rate of change is 30. (∆y = 30∆x). For each additional hour the castle is rented, the cost increases by an additional $30 b. Suppose he rents the bouncy castle, and at the last minute, the customer calls and wants to keep it for 3 more hours. How much more money will he have to pay? ∆c= 30(3) = $90 Note, that is $90 more dollars, added to what he was already paying. c. Write a formula to define the relationship between the total amount she charges (in dollars) in terms of the number of hours the castle is rented. (make sure to define your variables) Y = 30(x-2)+260  y = 30x + 200 d. In real life, what is the meaning of this formula? So, he charges a service fee of $200 no matter what and then $30 per hour to rent the castle. 28. a. Who is the fastest typer in Ms. Carlson’s typing class if John typed 250 words in 2 minutes, Katie typed 383 words in 3 minutes, and Jenny typed 1040 words in 8.5 minutes. Fastest -------------------------------------------------------------Slowest Jenny (122 WPM) John (125 WPM) Katie (128 WPM) b. Why is the unit rate useful in this situation? It allows us to easily compare their typing speeds. 29. Marc runs in a straight line from his house to the stoplight at the end of the street. a. Create the following graphs (make sure to add axis labels) Distance from home Distance from home with respect to time with respect to distance from stoplight b. Describe how each graph would change if Tom fell in the middle of his run (assume he gets back up and starts running again). The graph would flatten as time increased but distance does not. The graph would not change. 30. While skateboarding from one side of town to the other, Linda came across a huge dip in the road. a. Sketch a graph of her distance from the top of the hill (shown in picture) in terms of time since the top of the hill. b. *Challenge: Sketch a graph of her speed in terms of time since the top of the hill: