Semester 2 Final Exam Review Guide for AMS I

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1 Name: Semester 2 Final Exam Review Guide for AMS I Unit 4: Exponential Properties & Functions Lesson 1 Exponent Properties & Simplifying Radicals Products of Powers: when two powers with the same base are multiplied, you ADD exponents. b m b n = b m+n Power of a Power: when raising a power to an exponent, you multiply the exponents. (b m ) n = b m n Power of a Product: when raising a parenthesized product to an exponent, raise all factors in the parenthesis to that exponent. (ab) n = a n b n Power of a Quotient: when a fraction is raised to an exponent, distribute the exponent to both the numerator and denominator. ( a b )m = am Quotients of Powers: when dividing powers with the same base, SUBTRACT the exponent of the numerator minus the exponent of the denominator. bm b m b n = bm n Negative Exponents: move powers with negative exponents from the numerator to the denominator to make the exponent positive. Move negative exponents from denominator to the numerator. b n = 1 b n Zero Exponent: Any number raised to the zero power is 1. b 0 = 1 Square roots: The square root of a number is a value that, when multiplied by itself, gives the number. Example: 4 4 = 16, so a square root of 16 is 4. Radicals: expressions like b, 5, and 3 + b 2 are called radicals. Try these by simplifying each expression. No decimals or negative powers = = 3. (ab) 6 = 4. (b 2)3 = 5. ( 3 5 )3 = 6. y 4 y 5 =

2 = 8. ( 2 7 )3 = = 10. (xy)8 = = = Lesson 2 Exponential Growth & Decay Exponential growth is a situation that occurs when you are multiplying by a constant value larger than one. Equation is usually written in the form y = a (b) x, where a is the initial number and b is the constant multiplier. 13. Find a bank account balance if the account starts with $100, has an annual rate of 4%, and the money left in the account for 12 years. 14. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after How many cell phone subscribers were in Centerville in 1994? Exponential decay is a situation that occurs when an original amount is reduced by a consistent rate over a period of time. 15. You drink a beverage with 120 mg of caffeine. Each hour, the caffeine in your system decreases by about 12%. How long until you have 10mg of caffeine? 2

3 16. The amount of sunlight that reaches the plant life underwater determines the amount of photosynthesis that takes place. The table gives the percentage of sunlight that is present at various depths in a part of an ocean. Find an appropriate equation to model the data using regression. Round to the nearest tenth. Depth (meters) Percentage Of Light What percentage of light is present at a depth of 7 meters? Unit 5: Geometric Sequences & Logarithms Lesson 1 Logarithms & Solving Exponential Equations The exponent of a number says how many times you use the number in multiplication. In this example: 2 3 = = 8 (2 is used 3 times in a multiplication to get 8) A Logarithm goes the other way. A Logarithm says how many of one number to multiply to get another number. So a logarithm actually gives you the exponent as its answer: Definition of a Logarithm y= log b x if and only if b y =x Reads as y equals the log in base b of x Logarithms are specifically designed to help solve equations where the exponent is the variable. 18. Use the definition of a logarithm to rewrite each equation in exponential form (w/o logs) log 10 (100) = 2 log y = 4.5 log 7 x = 3 3

4 19. Use the definition of a logarithm to rewrite each equation in logarithmic form = 10,000 b 3 = = Apple juice has a hydrogen ion concentration of Find the ph of apple juice to the nearest tenth. Round to the nearest tenth. Use the formula ph = 1 log(h + ). 21. A very loud rock band played at a relative intensity of 125 db. What was the sound intensity in w/m 2? Write answer as a power of 10. Use the formula D = 10 log ( N 10 12). Steps in Solving Exponential Equations Isolate the power using order of operations. This means you want to get the base and the exponent alone on one side if the equation. Rewrite the equation from exponential form to logarithmic form. Solve for the unknown variable using order of operations. Example: = 10 x (subtract 1000 from both sides) = 10 x log = x (rewrite the equation in log form) = x (solve for x) 4

5 22. Solve each using logarithms. Round to the nearest hundredth when needed. a. 5(10) x 50 = 400 b. 10 x+2 = 1000 c. 3(10) x+4 = 3000 d. 10 2x = 50 Lesson 2 Geometric Sequences There are two types of sequences you should be aware of. The first is arithmetic sequences which increase/decrease by a constant value d. They are akin to linear functions. The second is geometric sequences which are multiplied by a constant value r. Each type of sequence can be written in explicit form or recursive form. The chart below explains what each looks like. Recall that you only need to know the first term and the d (or r) value to write/use any of these formulas. Explicit Form Arithmetic Sequence a n = a 1 + d(n 1) Recursive Form { a 1 = a n = a n 1 + d, n > 1 Geometric Sequence g n = g 1 r n 1 { g 1 = g n = r g n 1, n > 1 5

6 23. A house is worth $320,000 when purchased and was worth $335,000 after the first year and $350,000 after the second year. Assume the economy continues by this trend. a. Is the sequence arithmetic or geometric? Write an explicit formula and a recursive formula for the sequence. b. Determine the value of the house after 6 years. Show your work. 24. Records at Danbury Hospital show 2, 10, and 50 new cases of chicken pox were reported for 3 consecutive days. Assume that the number of new cases continues to increase at this rate. a. Is the sequence arithmetic or geometric? Write an explicit formula and a recursive formula for the sequence. b. How many new cases will be reported on the 5 th day? 6

7 UNIT 6 Probability Probability is the likelihood or chance that a given event will occur. Probability is usually expressed as a ratio of the number of likely outcomes compared with the total number of outcomes possible. P (event) = number of outcomes in the event number of possible outcomes 25. When different outcomes of an event are equally likely (for example, getting a head when you toss a coin), you can use a formula to calculate the probability of outcomes. When you roll a fair die there are six possible outcomes: 1, 2, 3, 4, 5, 6 a. What is the probability of getting an odd number? b. What is the probability of getting a number less than 4? c. What is the probability a 4 is rolled given the roll was even? d. What is the probability an even number is rolled given the roll is 3 or 4? Probability of Mutually Exclusive Events Two events are said to be mutually exclusive if they cannot happen at the same time. For example, if we toss a coin, either heads or tails might turn up, but not heads and tails at the same time. Similarly, in a single throw of a die, we can only have one number shown at the top face. The numbers on the face are mutually exclusive events. If A and B are mutually exclusive events then the probability of A happening OR the probability of B happening is P(A) + P(B). 7

8 P(A or B) = P(A B) = P(A) + P(B) When two events are NOT mutually exclusive then is an overlap between the events. P (A B)= P (A) + P (B) P(A B) If you roll a pair of 6 sided dice, find the probability that you get doubles or a sum of = **Do not count items twice** You need to subtract out the occurrence from where the overlap occurs. 26. Julia spins 2 spinners; one of which is labeled 1, 2 and 3, and the other is labeled 4, 5 and 6. Using the tree diagram for the experiment and list of possible outcomes. a) What is the probability that the spinners stop at 3 and 4? b) Find the probability that the spinners do not stop at 3 and 4. 8

9 c) What is the probability that the first spinner stops at 2 OR the sum is an even number? 27. Find the probability that a student plays lacrosse, given that the student is a junior. Class Freshman Sophomore Junior Senior Plays Lacrosse Does Not Play In an overpopulated country, a plan to limit growth was instituted. A family can have, at most, four children; however if there are two boys born before they reach the maximum of 4 kids, a family must stop having children. Will the population increase, decrease, or remain the same under this plan? Explain your reasoning. When using a random number generator on your calculator, you first need to enter in rand Int ( ). You then need to include parameters in the parentheses. The first number is your starting integer, the second number is your ending integer, and the third number is how many integers in the range will be chosen at a time. If you want to simulate rolling a 20 sided die five times: rand Int (1, 20, 5) 29. Use random generator to simulate spinning the spinner. 9

10 30. Rebecca is planning a birthday party and would like to have as many of her friends attend as possible. Her parents say that she can invite 30 people to her party. Rebecca knows that each person has an 80% chance of showing up. a. Describe one trial in a simulation of this situation using your calculator s random number generator. b. Rebecca conducted 50 trials using a random number generator. The results of how many people would show up to the party are displayed in the table below. Calculate the average number of people that will come to her party. Number of People Who Show Up Frequency Number of People Who Show Up Frequency Number of Trials Suppose Cracker Jack has 6 different prizes. Explain how to use the random integer function on your graphing calculator to continually buy boxes of Cracker Jack until you have received all of the prizes. Make sure to completely describe one simulation. 10

11 Unit 7 Nonlinear Functions (Quadratics) Function Notation: f(x) is read as f of x. It represents the same as the y value. 32. Evaluate f( 3) for the function f(x) = 7x 2 10x. Designing Parabolas Quadratic equation: y= ax 2 + bx + c If the a value is positive, it opens up If the a value is negative, it opens down The y intercept is (0, c) from the equation. All parabolas have a single vertical line that splits the graph in half. This line is called a line of symmetry. The line of symmetry will always go through the vertex or the minimum/maximum point. To find the minimum/maximum point, you take the average of the x-intercepts. 33. Consider the equation: f(x) = (x 8) (x 3) = x 2 11x + 24 a. What are the x intercepts? (, 0 ) (, 0) b. What is the y intercept? c. Does the graph have a minimum or maximum point? How can you tell? 11

12 d. Find the coordinates of the vertex. e. Graph the four points. 34. A parabola has a maximum point of (7, 12) and x-intercepts of 5 and 9. Determine the factored equation in the form f(x) = a(x? )(x? ). Expanding Binomials 35. Use FOIL method to expand factored form into standard form of ax 2 + bx + c. a. (x + 11)(x 12) b. (4 3x)(5x 8) Factoring Trinomials & Special Binomials When possible use Greatest Common Factor method first. Factor Differences of Squares such as x 2 a 2 = (x + a)(x a). Factor Trinomials with a 1 coefficient of x 2 term by: x 2 + bx + c = (x + )(x + ) x 2 bx + c = (x )(x ) x 2 bx c = (x + smaller)(x bigger) x 2 + bx c = (x + bigger)(x smaller) 12

13 If Trinomial has a coefficient of x 2 other than 1, use the big X method. a x c b value 36. Factor each binomial or trinomial. a. 6x 2 18x + 60 b. x x + 10 c. 49 x 2 d. 4x 2 25x 21 Solving Quadratic Equations Use basic order of operations to solve when there is an x 2 term but not an x term. Remember there are 2 answers. Example: 4x = 88 Use Greatest Common Factor method if x is present in all terms. Set each factor equal to zero to solve. Example: 6x 2 + 5x = 0 Use factoring methods described above problem 34 to factor and then set each factor equal to zero to solve. Example: x 2 + 6x + 8 = 0 13

14 37. Solve each quadratic equation using an appropriate method. a. 4x = 88 b. 6x 2 + 5x = 0 c. x 2 + 6x + 8 = 0 d. 2x 2 + 9x 35 = 0 Remember you may prepare a single 8 ½ X 11 sheet of paper with any notes, examples, rules, equations you would like to be used on the Final Exam. Study!!! 14