Tools:
Old Review (1-10)
1. Make a table with a repeat for x in the left column with 2 different values in the right or f(x) column
2. (0,3) is the y intercept, or b. m is -3/4. The equation, then is y = -3/4x + 3
3. The verticle line test determines if a graph represents a function by checking to see if there are 2 y values for the same x value. If so, the graph is NOT a function. It works because if you draw a vertical line through some x and it hits the graph in 2 places, that must be the second value for y given the same x.
4. The slope of a line is the rise/run, or how far you go over (run) on the bottom and how far you go up (rise) on the top
5. f(10) = 11 f(10) = 3/2(10) – 4 or 15-4 or 11
6. f(a) = h or f(age) = height
7. 0 to 110 or so.. the maximum age of a person
8. -11
9. 15(6)(52)
10. 7 evens for 15 sided dice, 3 odds for 6 sided dice, and 4 3’s in the deck. So.. 7(3)(4)
New (1-11)
1. Graph both lines. Where the 2 lines cross, that will be answer: the x value of this crossing point is the x value, and the y value will be the y value, assuming the lines are not the same or parallel.
2. y = -4x - 6 (use mathway.com)
3. f(x) = 650x + 500
4. 2x + 1 = x + 5 by substitution. So.. x + 1 = 5, so x = 4. Plug 4 into x for either original equation. I will pick the 2nd. y = (4) + 5, or y = 9. The solution is (4,5). This is where these 2 lines cross.
5. There is no solution
6. you change the inequality.. > becomes <, >= becomes <=, etc. For example, 4<5, but (-1)(4) > (-1)(5)
7. You want x all by itself in the middle. Get rid of the 1 by subtracting 1 from each side. -3 < x/3< 4 Now get rid of the bottom 3 by multiplying each piece by 3. -9 < x < 12
8. simplify 2(x-3) into 2x – 6, so you now have: 5-x<2x-6+5. This simplifies to: 5-x<2x-1. Combine x’s to get 5<x-1. Add 1 to both sides to get 6<x, or x>6 (same meaning) Since you never multiplies or divided by a negative number, the inequality sign does not change
9. f(x) = x: x<5
2x: x>=5
10. | x-8| must equal 6 or -6. so x could be 14 or 2
11. Your graph would go through (2,0) and this would be the bottom of the V. It would head up through (1,1) on the left and (3,1) on the right of the vertex (2,0)
PROBLEMS:
1. ANSWER: (10,-5,5) Solutions: x + y + z = 10; x + y = z; z = 3y + 2x These are the 3 equations that you are given in the text of the problem. Study it carefully to make sure you see where they come from and that you can do something similar another time. Now, pick one of the equations and solve it for one of the 3 varables. Then substitute this solved value into the other 2 equations. You now have 2 equations and 2 variables instead of 3 equations and 3 unknowns. Lets get that far. Notice that the second equation (x + y = z) is already solved for z. (as it the third equation) Lets pick the second and substitute (x + y) into the value for z in the first and third equations: You have: x+y+(x+y)=10 and x+y=3y+2x. Now we have 2 equations and just and x and a y. Where did the z go? We used the second equation to get rid of it, and put it into what z was, namely (x + y). Now, solve for x or y for one of these 2 remaining equations and substitute again. Lets simplify x+y+(x+y)=10 into 2x+2y=10. Lets solve for x (y would work too). x=(10-2y)/2. You can divide both pieces by 2 to simplify this into: x=5-y. Ok, now plug this in for x in the other equation, (x+y=3y+2x). You get: (5-y) + y = 3y + 2(5-y). Combine terms to get: 5=3y + 10 – 2y. This gets you: -5 = y. One down, 2 to go. Substitute 5 in for y in one of the 2 equations above: x+5=3(-5) + 2x (or you could use x+-5+(x+-5)=10) either way, x = 10. Now plug x = 10 and y = -5 into either of the original equations to find that z must be 5. Note that the 3 values (10,-5,5) will work for all 3 equations.
2. ANSWER: 266.6 square feet, $733.3 Solution: Painter A equation: y = 2x + 200 Painter B equation: y= 2.75x where x is the square footage painted and y is the total cost. Note that the 200 is the y intercept because you start out (with 0 for square footage) costing 200 dollars for Painter A. Substitute in for y to get: 2x + 200 = 2.75x. Now solve for x: 200 = .75x or x=200/.75 or x = 266.6 (approx). That means that if your house is over 266.6 square feet, you should choose Painter A, since after this square footage we only charges 2 dollars instead of 2.75.
3.The tolerance is .1, so anything between 9.1 and 9.2 centimeters (the design for the part is 9.2) is OK. The equation to represent this is: 9.1<=x<=9.3. To graph this, fill in the values between 9.1 and 9.3 on a number line.
4. f(s) = s<100,000: $5000
100,000<=s<=200,000: $5000 + s(.03)
s>200,000: $8000 + s(.05)
s = the sale price of the house, f(s) is the commission