2. Algebra Final Exam General Review

3. Find the domain and range. Is it a function? {(2,3),(-1,0),(2,-5),(0,-3)} Domain{2,-1,0} Range{3,0,-5,-3} Is it a function? Look for the same x value; if no repeats, then it is a function. If yes; do they have the same y value. If yes, it is a function. If no, not a function. The x values are the domain; the y values are the range. (2,3),(2,-5); the 2’s repeat; different y values; NOT a Function.

4. Practice: Find the domain and range of the following sets of ordered pairs. {(3,7),(-3,7),(7,-2),(-8,-5),(0,-1)} Domain:{3,-3,7,- 8,0} Range:{7,-2,-5,-1} The x values do not repeat; it is a function

5. Perfect Squares Some quadratic functions can written as a perfect square. x 2 + 8x + 16x 2 + 10x + 25 (x + 5) 2 (x + 4) 2 (x - 2) 2 (x - 6) 2 x 2 - 4x + 4x 2 - 12x + 36 Similarly when the coefficient of x is negative: The constant term is always (half the coefficient of x) 2.

6. Standard Form for Quadratic Equations Solve by Completing the square Add 7 to both sides. Coefficient of x is 6. Half of 6 squared is 9. Add it to both sides 99 Take the square root of each side Then solve for x. For Algebra: Both answers are correct, For Geometry only the positive answer is correct. (Solutions are roots.)

7. The roots are where the graph crosses the x axis y=x 2 -4 y=x 2 +2x-15 The roots: (2 and -2) (-5 and 3)

8. By completing the square on a general quadratic equation in standard form we come up with what is called the quadratic formula. This formula can be used to solve any quadratic equation whether it factors or not. If it factors, it is generally easier to factor---but this formula would give you the solutions as well. a=1; b=6; c=3 1 (1) 6 6 (3) Don't make a mistake with order of operations! Let's do the power and the multiplying first.

9. There's a 2 in common in the terms of the numerator Completing the square would have gotten the same solution (answer). NOTE: When using this formula if you've simplified and ended up with a negative inside the radical (square root sign), there are no real solutions. (There are complex (imaginary) solutions, but that will be dealt with in Calculus).

10. Fraction Rule (Addition and Subtraction) Unlike Denominators 3. 4.

11. Add or subtract rationals with unlike denominators The last two lines are both correct. The last line is simplified by factoring.

12. What is a reciprocal? Write the number as a fraction and flip it over. 6xy put a 1 under it and flip it over. 6xy becomes..1.. 1 6xy 3x becomes 2y 2y 3x

13. What are the x- and y- intercepts? The x-intercept is where the graph crosses the x-axis. The y-coordinate is always 0. The y-intercept is where the graph crosses the y-axis. The x-coordinate is always 0. (2, 0) (0, 6)

14. Find the x- and y-intercepts. 1. x - 2y = 12 x-intercept: Plug in 0 for y. x - 2(0) = 12 x = 12; (12, 0) y-intercept: Plug in 0 for x. 0 - 2y = 12 y = -6; (0, -6)

15. Exponent Examples

16. MULTIPLICATION PROPERTIES POWER TO A POWER This property is used to write and exponential expression as a single power of the base.

17. ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1. X 0 = 1

18. Solve systems (where they intersect) BY GRAPHING Y = 2X + 1 Y = -X + 4 (1,3) IS THE SOLUTION

19. Solve: x + y = 12 and -x + 3y = -8 by ELIMINATION x + y = 12 -x + 3y = -8 We need to eliminate (get rid of) a variable. The x’s will be the easiest. So, we will add the two equations. 4y = 4 Divide by 4 y = 1 Solve for x. Use equation 1 X+ y =12 y = 1; substitute x + 1 = 12 subtract 1 (each side) x = 11 (11,1) is solution Like variables must be lined under each other.

20. Work Problems Two people can do the same job but in different amounts of time, like 6 hours and 10 hours. How long would it take them together? Don’t average; it is NOT 8 hours!!! But each could do half the job in 3 and 5 hours So the answer is between 3 and 5. 6x + 10x = 60; 16x = 60; x = 3.75

21. Bart can wash the car in 20 min. Homer can wash it in 30 min. How long will it take them to wash the car if they both work together? The answer is between 10-15 min. 20x + 30x = 600; 50x = 600; x = 12 min

22. The 3 formulas for Speed, Time & Distance: Speed = Distance Time Time = Distance Speed Distance =Speed xTime Remember them from this triangle: D ST Solving for SpeedSolving for TimeSolving for Distance

23. 2 hour 30 mins Answer: He travelled 125 km A salesman travelled at an average speed of 50 km/h for 2 hours 30 minutes. How far did he travel? D ST Distance = Speed x Time = 50 x 25 = 125 km

24. Two cars leave the same town at the same time. One travels north at 60 mph and the other south at 45 mph. In how many hours will they be 420 miles apart? Bob Sherry DistanceTimeRate Car #1 Car #2 60 45 t t 60t 45t

25. Bob Sherry DistanceTimeRate Car #1 Car #2 60 45 t t 60t 45t In 4 hours the cars will be 420 miles apart. Opposite directions: add distances = miles.

26. At 10:00 A.M., a car leaves a house at a rate of 60 mi/h. At the same time, another car leaves the same house at a rate of 50 mi/h in the opposite direction. At what time will the car be 330 miles apart? DistanceTimeRate 1:00 P.M.

27. How much 20% alcohol solution and 50% alcohol solution must be mixed to get 12 gallons of 30% alcohol solution? Let x = amount of 20% 0.20x Let y = amount of 50% 0.50y want 12 gal of 30% 0.30(12) Counting Equation x + y = 12 Value Equation 0.2x + 0.5y =.30(12) y = 12 – x 0.2x + 0.5(12-x)=3.6 0.2x + 6 - 0.5x = 3.6 2.4 = 0.3x x=8; y=4 check: 0. 2(8) + 0.5(4) = 3.6 yes!

28. Example How many liters of 5% salt solution must I add to 2 liters of 2% salt solution in order to obtain a mixture that is 3.5% salt? Set up equation and solve for amounts. Use x to represent the liters of 5% salt solution

29. Amount of Salt Liters of Salt:.02(2) +.05(x) =.035(2 + x).04 +.05x =.070 +.035x 40 + 50x = 70 + 35x 15x = 30 x=2 liters of 5%.

30. A dairy has milk that is 4% butterfat and cream that is 40% butterfat. They want 36 gal that is 20% butterfat, how many gallons of milk and cream must be added. Set up equations and solve. Answer: mix together…… 20 gallons of milk and 16 gallons of cream

31. Solving Abs Value Equations With absolute value signs, we must assume that the answer can be both positive and negative because what is inside the absolute value sign can be either positive or negative. Solve | x – 4 | = 8 | x – 4 | is positive | x – 4 | is negative | x – 4 | = 8 x – 4 = 8 x – 4 = -8 Add 4 to each side The equation has TWO solutions. Check BOTH. x = 12x = -4 OR Original Problem

32. Solving Absolute Value Equations Solve | 2x – 7 | - 5 = 4 Hint: Simplify FIRST. Why? | 2x – 7 | is positive | 2x – 7 | is negative | 2x – 7 | - 5 = 4 2x - 7 = 92x – 7 = -9 Add 5 to each side The equation has TWO solutions. Check BOTH. | 2x – 7 | = 9 add 7 2x = 162x = -2 x = 8x = -1 Problem Absolute value is a grouping symbol. ISOLATE IT FIRST.

33. What is a projectile? A projectile is any object which once projected or dropped continues in motion by its own inertia and is influenced (changed) only by the downward force (pull) of gravity.

34. Height = - 4.9t 2 + 78.4 1. Anna drops a ball from rest from the top of 78.4-meter high cliff. How much time will it take for the ball to reach the ground ? At what height will the ball be after 3 seconds of motion?

35. Height = - 4.9t 2 + 78.4 How much time will it take for the ball to reach the ground ? 0 = - 4.9t 2 + 78.4 At what height will the ball be after 3 seconds of motion? Height = - 4.9(3) 2 + 78.4

36. BUILDING UNDERSTANDING FOR THE EQUATION USED TO REPRESENTS THE MOTION (TRAJECTORY) OF A ROCKET

37. A model rocket blasts off. Its velocity at that time is 50 m/s. Assume that it travels straight up and that the only force acting on it is the downward pull of gravity. In the metric system, the acceleration due to gravity is 9.8 m/s 2. The quadratic function h(t)=(1/2)(-9.8)t 2 +50t describes the rocket’s projectile motion as a function of time, t. Height = -4.9t 2 + 50t Graph the quadratic to see its path, height at x sec.

38. BUILDING UNDERSTANDING THIS IS THE SAME AS THE ROCKET EXCEPT THAT THE BALL IS HIT (STARTS) AT 3 FEET ABOVE THE GROUND.

39. A baseball batter pops a ball straight up. The height of the ball is shown as a function of time in a graph. What types of information can be learn about the height of the ball?

40. The ball's initial height when hit by the bat (0,3) Hits the ground in just over 4 seconds. The maximum height almost 70 ft in 2 sec (2,68) How many times is the ball 20 feet above the ground? Twice. At about 0.3 sec and 3.6 sec.