1.1 The Language of Algebra 1. What does the term variable mean?

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1 Advanced Algebra Chapter 1 - Note Taking Guidelines Complete each Now try problem after studying the corresponding example from the reading 1.1 The Language of Algebra 1. What does the term variable mean? 2. Define algebraic expression. 3. What is an algebraic sentence? 4. Study example 1 5. Now try the following problem: a. Express the cost of w pencils at b cents per pencil. 6. Study example 2 7. Now try the following problem: a. Eden has $75 in the bank. If she saves $25 each month, how much money, excluding interest, will she have in the bank after m months?

2 8. What is meant by evaluating an express? 9. Rules for Order of Operations 10. Define equation. 11. Define Formula 12. Study example Now try the following problem: 4 3 a. Use the formula V = π r to find the 3 volume of a ball that is 10 centimeters in diameter. Summarize what you learned in sections 1.1:

3 1.2 What is a Function? 1. How do you identify the dependent variable in an algebraic situation? 2. How do you identify the independent variable in an algebraic situation? 3. Definition of a Function: 4. Statement of a function relation: The dependent variable is a function of the independent variable. OR Is the dependent variable a function of the independent variable? 5. Study example 1 6. Now try the following problem: a. The table below shows the average temperature T in degrees Fahrenheit for each month M in Honolulu, Hawaii. M T M T Jan. 73 July 80 Feb. 73 Aug. 81 Mar. 74 Sept. 81 Apr. 76 Oct. 80 May 78 Nov. 77 June 79 Dec. 74 i. Is T a function of M? Justify. (What is the generic ordered pair?) ii. Is M a function of T? Justify. (What is the generic ordered pair?)

4 7. Define the domain of a function. 8. Define the range of a function. 9. Study example Now try the following problem: a. Give the domain and range for the situation in #6i on the previous page. 11. Define: a. Natural numbers: b. Whole numbers: c. Integers: d. Rational numbers: e. Real numbers: f. Irrational numbers: 12. Study example Now try the following problem: a. The total cost c of v cans of vegetable soup at 59 cents per can is given by the formula c = 59v. (What is the generic ordered pair?) i. Is c a function of v? Why or why not? ii. What are the domain and range of this function? Summarize what you learned in sections 1.2:

5 1.3 The Function Notations 1. Euler Notation or f(x) Notation: 2. Study example 1 3. Now try the following problem: a. If f(x) = x 2 2x, find f(-7) 4. Study example 3a 5. Now try the following problem: a. The area of a circle is a function of its radius. Rewrite the formula A = π r 2 i. Using Euler s notation Summarize what you learned in sections 1.3:

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7 1-4 Graphs of Functions 1. Study example 1 2. Now try the following problem a. The graph below gives the distances a person traveled away from home during a 7-hour trip. i. Estimate the distance away from home after 1 hour. ii. Estimate how long it took to get 75 miles away from home. iii. Identify the independent and dependent variables. iv. State the domain and range of the function 3. What does the term relation mean? 4. Definition of a Function:

8 5. Study example 2 6. Now try the following problem a. Consider the graph of a line that passes through (0, -3) and is parallel to the x-axis. Is this the graph of a function? Justify your answer. 7. What do you know about the points on a vertical line? 8. Theorem Vertical-Line Test for Functions 9. What do you know about the points that make up a function? Summarize what you learned in sections 1.4:

9 1-5 Solving Equations 1. Give two key questions that arise when working with Functions. 2. Study example 1 3. Now try the following problem Erica just bought a used car with 33,000 miles on it. She expects to drive it about 800 miles/month. If m = the number of months Erica has had the car, then f(m) = 33, m is the expected total number of miles the car will have been driven after m months. a. How many miles will be on the car after a year and a half? b. Estimate when the car will have been driven 100,000 miles. 4. Distributive Property: 5. Study example 2 6. Now try the following problem a. An individual put total of $500 into two accounts. One paid 5.5% annual interest, and the other paid 4% annual interest. How much was invested in each account if the total yearly interest was $21.88?

10 7. What does the process of clearing fractions involve? 8. Study example 3 9. Now try the following problem a. A concert group plans to visit 3 cities on a tour. One third of the total distance to be traveled by the group is from its home city to the first city. One fifth of the total distance is from the first city to the second city. It is 273 miles from the second city to the third city. Find the total number of miles traveled. 10. Opposite of a Sum Theorem Summarize what you learned in sections 1.5:

11 1-6 Rewriting Formulas 1. When a formula is solved for a variable: 2. What does the phrase ïn terms of refer to? 3. Study example 1 4. Now try the following problem a. K = C relates degrees Kelvin to degrees Celsius. Solve for C. 5. Study example 2 6. Now try the following problem a. The following formula gives the final velocity v for an object with initial velocity vo, which accelerates at a uniform rate of acceleration a for t seconds. v = vo + at i. Solve this formula for a. ii. Solve this formula for t.

12 7. Study example 3 8. Now try the following problem a. In a right triangle, the product of the legs a and b equals what quantity? 9. What does the vocabulary word pitch mean? Summarize what you learned in sections 1.6:

13 1-7 Explicit Formulas for Sequences 1. Sequence 2. Term of a Sequence 3. Explicit Formula 4. Study example 1 5. Now try the following problem a. Use n(n + 1) to give the 15 th rectangular number. 6. Subscript 7. Subscripted Variable 8. Index

14 9. Generate the terms of a sequence 10. Study example Now try the following problem a. Consider the formula tn = (n 1) for integers n >= 1. i. What are the first four terms of the sequence generated by the formula? ii. Find t Study example Now try the following problem a. Suppose you drop a ball from the top of a 50-foot wall, and, on each bounce, the ball rises to 75% of its previous height. The heights of the ball after each bounce form a sequence. i. Write the first three terms of the sequence. ii. After how many bounces will the ball rise less than 10 feet? 14. Now try the following problem a. Consider the sequence, t, of squares of consecutive positive integers. i. What is the value of t4? ii. Give an explicit formula for tn. iii. What is the value of t250 Summarize what you learned in sections 1.7:

15 1-8 Recursive Formulas for Sequences 1. Thinking recursively: 2. Recursive Formula 3. Study example 1 4. Now try the following problem a. Consider the sequence defined by the recursive formula and generate the first six terms. S1 = 1 s n = 3 previous term -1 for integers n 2 5. What is the brace used for in a recursive formula? 6. Recursive definition 7. Ans key

16 8. Study example 2 9. Now try the following problem a. Consider the sequence defined by the recursive formula and generate the first six terms. T1 = 10 Tn = ANS + 6 for integers n Study example Now try the following problem a. Louis is trying to learn a very long and difficult piano piece that has 400 measures. After learning the first page, which has 28 measures, he decided that he would learn 4 measures a day. The sequence 28, 32, 36, 40, gives the number of measures he will have learned after n days. i. In words, describe this sequence recursively. ii. Write a recursive formula for this sequence using the ANS key. 12. Now try the following problem a. Write the calculator key sequence that generates the sequence 1, 4, 7, 10, 13, Summarize what you learned in sections 1.8:

17 1-9 Notation for Recursive Formulas 1. Study example 1 2. Now try the following problem a. Consider the sequence defined by the recursive formula: t1 = 3 tn = 2tn-1 + 1, for integers n 2 i. Describe the sequence in words. ii. Find the first four terms of the sequence. 3. Study example 2 4. Now try the following problem a. Find v1, v2, v3, v4, and v5 when v1 = 15 vn = vn-1-4n + 2, for integers n 2

18 5. Study example 3 6. Now try the following problem a. Write the first eight terms of the sequence that is generated by the following formula: = = = 3 for integers n a a a a a n n n 7. Fibonacci sequence: Summarize what you learned in sections 1.9: