1. Pre-Algebra Patterns and Sequences What is the common difference in the sequence –6, –3, 0, 3,... ? Lesson 13-1 The common difference is 3. –6 –3 0 3 +3 +3 +3 Find the common difference. Additional Examples

2. Pre-Algebra Patterns and Sequences Lesson 13-1 A swimmer training for a meet swims 5 laps the first day, 6 laps the next day, 8 laps the third day, and so on. Find the next three terms of the sequence. Then write a rule to describe the sequence. 1212 5 6 8 1212 The next three terms are 9, 11, and 12. 1212 1212 The rule for the sequence is Start with 5 and add 1 repeatedly. 1212 +1 1212 1212 Find the common difference. 9 11 12 1212 1212 +1 +1 +1 1212 1212 1212 Use it to find the next three terms. Additional Examples

3. Pre-Algebra Patterns and Sequences Find the common ratio in the sequence 3, 9, 27, 81,... Find the next three terms of the sequence. Then write a rule to describe the sequence. Lesson 13-1 3 9 27 81 The common ratio is 3, and the next three terms are 243, 729, and 2,187. The rule for the sequence is Start with 3 and multiply by 3 repeatedly.  3  3  3 Find the common ratio. 243 729 2,187  3  3  3 Use it to find the next three terms. Additional Examples

4. Pre-Algebra Patterns and Sequences Tell whether each sequence is arithmetic, geometric, or neither. Find the next three terms of each sequence. Lesson 13-1 a. 3, 5, 9, 15,... The sequence is neither arithmetic nor geometric. Following the pattern above, the next three terms are 23, 33, and 45. 3 5 9 15 b. 12, 10.5, 9, 7.5,... There is a common difference of –1.5. The next three terms are 6, 4.5, and 3. 12 10.5 9 7.5 The sequence is arithmetic. +2 +4 +6 23 33 45 +8 +10 +12 –1.5 –1.5 –1.5 6 4.5 3 –1.5 –1.5 –1.5 Additional Examples

5. Pre-Algebra Patterns and Sequences (continued) Lesson 13-1 c. 2, –4, 8, –16,... 2 –4 8 –16 The sequence is geometric. The ratios for the first four terms are,, and. These equal –2, which is the common ratio.  –2  –2  –2 –4 2 –16 8 –4 32 –64 128  –2  –2  –2 The next three terms are 32, –64, and 128. Additional Examples

6. Pre-Algebra Graphing Nonlinear Functions For the function y = –x 2 + 1, make a table with integer values of x from –2 to 2. Then graph the function. Lesson 13-2 x –x 2 + 1 = y (x, y) –2 –(–2) 2 + 1 = –3 (–2, –3) –1 –(–1) 2 + 1 = 0 (–1, 0) 0 –(0) 2 + 1 = 1 (0, 1) 1 –(1) 2 + 1 = 0 (1, 0) 2 –(2) 2 + 1 = –3 (2, –3) Make a table.Make a graph. Additional Examples

7. Pre-Algebra Graphing Nonlinear Functions Lesson 13-2 The function A = 12x – x 2, where x is width in meters, gives the area A of a garden in square meters. Graph the function. Use the graph to find the width that gives the greatest area. x 12x – x 2 = y (x, y) 0 12(0) – (0) 2 = 0 (0, 0) 1 12(1) – (1) 2 = 11 (1, 11) 2 12(2) – (2) 2 = 20 (2, 20) 3 12(3) – (3) 2 = 27 (3, 27) 4 12(4) – (4) 2 = 32 (4, 32) 5 12(5) – (5) 2 = 35 (5, 35) 6 12(6) – (6) 2 = 36 (6, 36) 7 12(7) – (7) 2 = 35 (7, 35) 8 12(8) – (8) 2 = 32 (8, 32) The ordered pair (6, 36) shows what appears to be the highest point. So the width 6 meters gives the greatest area. Additional Examples

8. Pre-Algebra Graphing Nonlinear Functions Graph the function y = |x| – 1. Lesson 13-2 x |x| – 1 = y (x, y) –2 |–2| – 1 = 1 (–2, 1) –1 |–1| – 1 = 0 (–1, 0) 0 |0| – 1 = –1 (0, –1) 1 |1| – 1 = 0 (1, 0) 2 |2| – 1 = 1 (2, 1) Additional Examples

9. Pre-Algebra Exponential Growth and Decay For the function y = 4 x, make a table with integer values of x from 1 to 4. Then graph the function. Lesson 13-3 x 4 x y (x, y) 1 4 1 4 (1, 4) 2 4 2 16 (2, 16) 3 4 3 64 (3, 64) 4 4 4 256 (4, 256) Additional Examples

10. Pre-Algebra Exponential Growth and Decay Lesson 13-3 For y = 4(2) x, make a table with integer values of x from 0 to 4. Then graph the function. x 4(2) x y (x, y) 0 4(2) 0 4 (0, 4) 1 4(2) 1 8 (1, 8) 2 4(2) 2 16 (2, 16) 3 4(2) 3 32 (3, 32) 4 4(2) 4 64 (4, 64) Additional Examples

11. Pre-Algebra Exponential Growth and Decay For the function y = 2(0.5) x, make a table with integer values of x from 0 to 5. Then graph the function. Lesson 13-3 x 2(0.5) x y (x, y) 0 2(0.5) 0 2 (0, 2) 1 2(0.5) 1 1 (1, 1) 2 2(0.5) 2 0.5 (2, 0.5) 3 2(0.5) 3 0.25 (3, 0.25) 4 2(0.5) 4 0.125 (4, 0.125) 5 2(0.5) 5 0.0625 (5, 0.0625) Additional Examples

12. Pre-Algebra Polynomials Is each expression a monomial? Explain. Lesson 13-4 a. 5 + c c. 6ab 2 No; the expression is a sum. Yes; the expression is the product of the real number 6 and the variables a, b, and b. No; the denominator contains a variable. b. 7z 3 d. 4g h4g h Yes; the expression is the product of the real number and the variable z. 7373 Additional Examples

13. Pre-Algebra Polynomials Lesson 13-4 Tell whether each polynomial is a monomial, a binomial, or a trinomial. a. 14x 2 + 2xy – 7y 2 c. z + 10 b. 11a 2 bc 3 binomial monomial trinomial Additional Examples

14. Pre-Algebra Polynomials Evaluate each polynomial for r = 2 and s = 7. Lesson 13-4 a. 5r 2 – s 5r 2 – s = 5(2) 2 – 7Replace r with 2 and s with 7. = 13Simplify. b. 6rs 3 = Replace r with 2 and s with 7. 6rs 3 6(2)(7) 3 = 28Simplify. Additional Examples

15. Pre-Algebra Polynomials The polynomial –16t 2 + 100t gives the height, in feet, reached by a fireworks shell in t seconds. If the shell explodes 5 seconds after launch, at what height did it explode? Lesson 13-4 –16t 2 + 100t –16(5) 2 + 100(5)Replace t with 5. 100Simplify. The shell exploded at 100 feet. Additional Examples

16. Pre-Algebra Adding and Subtracting Polynomials Simplify (4b 2 + 2b + 1) + (7b 2 + b – 3). Lesson 13-5 Method 1: Add using tiles. 4b 2 + 2b + 1 7b 2 + b – 3 The sum is 11b 2 + 3b – 2 Additional Examples

17. Pre-Algebra Adding and Subtracting Polynomials (continued) Lesson 13-5 Method 2: Add by combining like terms. (4b 2 + 2b + 1) + (7b 2 + b – 3) = (4b 2 + 7b 2 ) + (2b + b) + (1 – 3)Use the Commutative and Associative Properties of Addition to group like terms. = (4 + 7)b 2 + (2 + 1)b + (1 – 3)Use the Distributive Property to combine like terms. = 11b 2 + 3b – 2Simplify. Additional Examples

18. Pre-Algebra Adding and Subtracting Polynomials Lesson 13-5 Find the sum of 2z 2 – 9z – 15 and 8z + 11. 2z 2 – 9z – 15 8z + 11 Align like terms. + 2z 2 – z – 4 Add the terms in each column. Additional Examples

19. Pre-Algebra Adding and Subtracting Polynomials Simplify (12y 2 + 10y –5) – (6y 2 + 8y – 11). Lesson 13-5 Additional Examples (12y 2 + 10y – 5) – (6y 2 + 8y – 11) = 12y 2 + 10y – 5 – 6y 2 – 8y + 11 Write the opposite of each term in the second polynomial. = (12y 2 – 6y 2 ) + (10y – 8y) + (–5 + 11) Group like terms. = (12 – 6)y 2 + (10 – 8)y + (–5 + 11) Use the Distributive Property. = 6y 2 + 2y + 6 Simplify.

20. Pre-Algebra Multiplying a Polynomial by a Monomial Find the area of the rectangle. All measurements are in meters. Lesson 13-6 The area of the rectangle is (4v 2 + 28v) m 2. A = w = 4v(v + 7)Substitute. = 4v(v) + 4v(7)Use the Distributive Property. = 4v 2 + 28vSimplify. Additional Examples

21. Pre-Algebra Multiplying a Polynomial by a Monomial Lesson 13-6 Simplify 5n 2 (2n 3 – 4n 2 + n). 5n 2 (2n 3 – 4n 2 + n) Additional Examples = 5n 2 (2n 3 ) + 5n 2 (–4n 2 ) + 5n 2 (n) Use the Distributive Property. = (5)(2)n 2 + 3 + (5)(–4)n 2 + 2 + (5)n 2 + 1 Use the Commutative Property of Multiplication. = (5)(2)n 5 + (5)(–4)n 4 + (5)n 3 Add exponents. = 10n 5 – 20n 4 + 5n 3 Simplify.

22. Pre-Algebra Multiplying a Polynomial by a Monomial Write 6r 4 + 10r 3 – 14r 2 as a product of two factors. Lesson 13-6 6r 4 = 2 3 r r r r 10r 3 = 2 5 r r r Write prime factorizations. –14r 2 = –1 2 7 r r GCF = 2r 2 Find the GCF. Write each term as the product of 2r 2 and another factor. 6r 4 = 2r 2 3r 2 10r 3 = 2r 2 5r –14r 2 = 2r 2 –7 6r 4 + 10r 3 – 14r 2 = 2r 2 (3r 2 + 5r – 7)Use the Distributive Property. Additional Examples

23. Pre-Algebra Multiplying Binomials Simplify (x + 3)(x + 5). Lesson 13-7 The area is x 2 + 8x + 15. Additional Examples

24. Pre-Algebra Multiplying Binomials Lesson 13-7 Simplify (b + 2)(3b – 1). (b + 2)(3b – 1) = b(3b – 1) + 2(3b – 1)Use the Distributive Property. = 3b 2 – b + 6b – 2Use the Distributive Property again! = 3b 2 + 5b – 2Simplify. Additional Examples