You can download the quizzes themselves on the Resources link on the website. Jump to week 4. Use the quizzes to review the material on a regular basis.
TOOLS quiz: 18 questions, 40 points
1. Pick an ordered pair where you have the same first value (x) but diferent second values. This cannot be a function. Example: (2,3), (3,4), (2,5)… x=2 has 2 different values, so NOT a function
2. Since slope = -3, m = -3. The equation to start with is: y = -3x + b. Now solve for b. Take the point you have, (2,5) and plug 2 in for the x and 5 in for the y. This gives you: 5 = -3(2) + b. Solve for b. b=11. Therefore, the equation is: y = -3b + 11
3. The vertical line test is used to check if a graph represents a function. It works because if you can draw a vertical line through any x value and have the vertical line pass through 2 different points, it cannot be a function graph.
4. They must be parallel, since they have the same slope but have different y intercepts
5. p = f(h)
6. Since there are 24 hours in a day, and the x axis is hours, the practical domain is 0-24
7. First find m. The change in x is from 4 to 6, or 2. The change in y is from 5 to 4 or -1. The slope, then, is -1/2. The equation so far, then, is y=-1/2x + b. Solve for b by taking one of the points (I will take the first one, (4,5)) and plug in the x and y values into this first equation and solve for b. 5 = -1/2(4) + b, 7 = b. The equation is: y=-1/2x + 7
8. After multiplying and dividing from left to right you have: -4 + 6 – (-2) + (-6) + (-3). Combine left to right and you get: -5
9. There are 3 ways to get an even number on the first dice. There are 2 ways to get a number less than 3 on the blue dice. There are 12 face cards in the deck. The total combinations, then, would be 3(2)(12) or 72
10. If you have a graph of 2 lines and they cross at a point, lets say (3,1), that means that when x = 3 for both functions you have the same y value. If you are trying to solve for the x value where the 2 functions have the same value, the are trying to find this x value.
11. Your graph would have a point at (0,-2). this would be the bottom on the V of the graph. 2 other points on your graph would be (-1,1) and (-3,1)
12. (4 points) Combine all like terms together: 2x + 3y/2 -4 = 4x + y – 2. Get the y terms on one side and the x and numbers on the other: 3y/2 – y = 2x + 2. You have several choices here, but one way now is to multiply both sides by 2 so you can get rid of the 2 on the bottom os 3y/2. So…. 2(3y/2 – y) = 2(2x + 2) or 6y/2 – 2y = 4x + 4 or 3y – 2y = 4x + 4. That reduces to your answer: y = 4x + 4
13. y = 120x + 100 The y intercept is 100, since you start with 100 minutes. THe slope is 120, since you get 120 free minutes every month. y = free minutes, x = months you have the plan
14. DROPPED: bad and unclear question. Should have been: what happens to x > y when you multiply both sides by (-1). Answer: -x < -y
15. 14 -21x + 14y
16. (4 points) 5 < (3/2)x + 2 < 11 Subtract 2 from all parts of this equation to get: 3 < (3/2)x < 9. Several choices now.. Lets multiply each piece by 2 so you can get rid of the 2 on the bottom of the fraction, since you want x all by itself. You get: 3(2) < 2(3/2)x < 2(9) or 6 < 3x < 18. Get rid of the 3 to get your answer by dividing by 3 to get rid of the number next to the x: 2 < x < 6
17. Your tolerance is .05, therefore your part cannot be bigger or smaller than what is is supposed to be, plus or minus .05 millimeters. Since the part is designed to be 120.2 millimeters, add .05 to find out how big it could be to be within the tolerance, and subtract .05 from 120.2 to find how small it could be. Your answer, then is: 120.15 < x < 120.25
18. A piecewise function is a function that has a different formula to find the y value for different ranges of x values.
PROBLEM SOLVING ANSWERS: 20 points, 10 points each. Partial credit given, so show your work
1. The sum of 3 numbers is 18. The first number plus the second number equals the third number. Twice the first number plus the third number equals twice the second number minus 1. Find each number
Solution: The 3 equations are: x + y + z = 18; x + y = z; and 2x + z = 2y-1. If you have trouble seeing where these equations come from, get help right away. Later we will use real world examples where finding the equations may be a bit harder. OK, you have 9 things you could do at this point, but the idea is to pick one of the equations, solve for one of the variables, and plug this answer in for that variable in the other 2 equations. This will effectively change a 3 equations 3 variables problem into a 2 equations, 2 variables problem. If you look at the 2nd equation, it is already solved for z… that is z = x + y. so, I can replace z with (x+y) every time I see it in the first and last equation. Then I can just forget about this 2nd equation. It has served its purpose. You get this:
x + y + (x+y) = 18 (plugging in for the z in the first initial equation) and 2x + (x+y) = 2y-1 (plugging in for the z in the third equation). The work of the 2nd initial equation is over. Cross it off. Now combine terms in both equations: 2x + 2y = 18, and 3x + y = 2y – 1. Now, you do the same thing again, only with 2 equations.. that is, use one equation to solve for a variable and plug your anser into the remaining equation. Which equation and variable to pick?? You have 4 choices. All will give you the same answer. I will solve for y in the second equation. 3x + y = 2y – 1. Get the ys on one side and the other stuff on the other.. 3x + 1 = y (add 1 to both sides then subtract y)… Now I have solved for y. Take the other equation and replace every y with (3x+1) This equation starts out as: 2x + 2y = 18, but after the substitution you get: 2x + 2(3x + 1) = 18. Now solve for x. Distribute to get: 2x + 6x + 2 = 18. Combine to get: 8x + 2 = 18. Solve for x: x = 2. So now you know the x value, and you can replace x with 2 in one of your 2 equations above, that you got after your initial substitution (2x + 2y = 18 and 3x + y = 2y – 1) I will use the second, just for kicks.. Since x = 2, substitute to get: 3(2) + y = 2y – 1. Solve for y to get: y = 7. Now pick any of the original equations to solve for z.. I will use the first (x + y + z = 18), 2 + 7 + z = 18. z must be 9. SO….. your final answer is:
(2,7,9) Notice that these values work for all 3 equations
2. DirecTV has a lifetime plan that costs $150 to join, but then will only charge $35 per month for the life of your account for their TV package. A web based company has a similar offer, but they require only $70 up front, and then charge $38 per month for the same package. How many months will it be before DirecTV is a better deal?
Solution: This problem is VERY similar to your first problem, except you will START with 2 equations and 2 unknowns or variables. The directv plan costs you 150 to join, so you start out paying 150 dollars, even when the x value, in this case the months, is 0. So the b is 150. You know that for every month you go over, you have to pay $35 (over 1 month, up 35 dollars), so your slope is 35. Your equation is: y = 35x + 150. In a similar way, the web offer equation is: y = 38x + 70. You now have 2 equations and 2 unknowns (or variables). Just like the above problem, you can substitute one solved value (y in this case) into the other equation.. This gives you 35x + 150 = 38x + 70 (it does not matter in this case which equations you substitute, you get to the same place). Now solve for x: 80 = 3x, or x = 26.66 months. So, when a little less than 27 months go by, the 2 plans will cost you the same amount of money. So after this time, Directv is a better deal.
answer: 26.66 months