1. REVIEW: SLOPE OF A LINE What is slope of a line? --how to move from one point on the line to another point on the line A B C D --these moves are classified as rise (up-down) move over run(right-left) move run 3 rise 2 run 3 rise 2 run 3 rise 2 The slope between any two points on the same line is always the same.

2. REVIEW: SLOPE OF A LINE What is slope of a line? --The coordinates of two points on a line can be used to find the slope of the line by the formula: A B C D (- 2, 0) (1, 2) (4, 4) (7, 6) EXAMPLE: (using points A and B) rise run

3. 5, 7, 9, 11, 13, 15, 17 +2 USING SLOPE: NEXT/NOW STATEMENTS Consider the following number sequence: Value (y) change Position (x): 0 123 4 5 6 Starting Position +1 Starting Value

4. USING SLOPE: NEXT/NOW STATEMENTS 12 3 4 56 5 7 9 11 13 15 17 0 SLOPE-INTERCEPT EQUATION OF A LINE: y = m x + b slope In this problem: y = 2 x + 5 y- Intercept slope y- Intercept (starting value)

5. USING SLOPE: NEXT/NOW STATEMENTS 12 3 4 56 5 7 9 11 13 15 17 0 SLOPE-INTERCEPT EQUATION OF A LINE: y = slope (x) + y- intercept Another way to compute the value (y): NEXT (y value) = NOW (present y value) + SLOPE; STARTING AT Y-INTERCEPT

6. USING SLOPE: NEXT/NOW STATEMENTS 12 3 4 56 5 7 9 11 13 15 17 0 NEXT (y value) = NOW (present y value) + SLOPE; STARTING AT Y-INTERCEPT EXAMPLE: Starting at a y- value of 9 (NOW) Add the slope (2) 9 + 2 = 11 which is the NEXT y-value

7. Initial Value 1st2nd3rd4th There are two inches on the ground at midnight. Each hour the snowfall increase by ½ inch. SITUATION 1 Snowfall Xy TABLE GRAPH SEQUENCE SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENTPractical Domain Practical Range EQUATION 2 1/21/2 Y = 1 / 2 x + 2 Hour after midnight Snowfall (inches) 0 1 2 3 2.5 2 3 3.5 +1/2+1/2 +1/2+1/2 2 +1/2+1/2 +1/2+1/2 +1/2+1/2 +1/2+1/2 2.533.54 NEXT = NOW + ½; starting at 2 All non-negative numbers All numbers at least two 1 2 1 2 +1/2+1/2

8. There are 6 inches of rain on the ground at midnight. It keeps falling at ¼ inch per hour. SITUATION 2 Rainfall Xy TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENT Practical Domain Practical Range EQUATION y = ¼ x + 6 ¼ 6 Hours after midnight rainfall (inches) NEXT = NOW + ¼ ; STARTING AT 6 06 1 6.25 26.50 3 6.75 47.00 6 6.256.50 6.75 7.00 x ≥ 0 y ≥ 6

9. SITUATION 3 (Cab fare) X Miles Y Cab fare 03 15.50 28 310.5 TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENT Practical Domain Practical Range EQUATION A cab company charges a $ 3 flat rate plus $ 2.50 per mile. 2.5 3 y = 2.5 x + 3 3 5.508 10.50 13 NEXT = NOW + 2.5; STARTING AT 3 x ≥ 0 y ≥ 3

10. SITUATION 4 (money saved in bank) Xy TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th 507090110 SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENT Practical Domain Practical Range EQUATION You have a beginning balance of $ 30. You add $ 20 every week. Weeks of saving Amount 20 30 y = 20 x + 30 030 1 50 270 3 90 4 110 30 NEXT = NOW + 20; STARTING AT 30 x ≥ 0 y ≥ 30

11. SITUATION 5 (money in wallet paying for sodas) Xy TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th 8642 SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENT Practical Domain Practical Range EQUATION John has $ 10 in his wallet. Each soda he buys costs $ 2.00. - 2 10 Number of sodas bought Amount left 010 1 8 2 6 3 4 4 2 y = - 2 x + 10 01 2 3 4 5 6 7 8 910 11 1 2 3 4 5 6 7 8 10 11 9 NEXT = NOW – 2 ; STARTING AT 10 0 ≤ x ≤ 5 0 ≤ y ≤ 10

12. SITUATION 6 (Water in bathtub) Xy TABLE GRAPH SEQUENCE SLOPE (rate of change) NEXT-NOW STATEMENT Practical Domain Practical Range EQUATION 4 - 2 Slope = - 2 / 4 = - ½ A bathtub has 2 gallons of water until the plug is pulled. It then loses ½ gallon per minute. - ½ Initial Value 1st2nd3rd4th Y-intercept (initial value) 2 Number of hours after Amount left 02 11.5 21.0 30.5 4 0 y = - ½ x + 2 2 1.51.00.50 NEXT = NOW – ½ ; STARTING AT 2 0 ≤ x ≤ 4 0 ≤ y ≤ 2

13. SITUATION 7 (Filling a bathtub) Xy TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENTPractical Domain Practical Range EQUATION Next = Now + ½; starting at 3 A bathtub has 3 gallons of water. The water is turned on and it fills at a rate of ½ gallon per minute. ½ 3 Number of hours after Amount 03 13.5 24.0 34.5 45.0 y = ½ x + 3 3 3.54.04.55.0 x ≥ 0 y ≥ 3

14. SITUATION 8 X Time (min) Y Words typed 248 496 TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENT Practical Domain Practical Range EQUATION Slope = (96 – 48) / (4 – 2) = 48 / 2 = 24 Sue can type 24 words per minute. 24 0 00 1 372 0 244872 96 y = 24 x NEXT = NOW + 24 ; STARTING AT 0 x ≥ 0 y ≥ 0

15. SITUATION 9 (Lemonade Stand) Xy TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th SLOPE 2 (rate of change) Y-intercept -10 (initial value) NEXT-NOW STATEMENTPractical Domain Practical Range EQUATION 0 1 234 5 67 8 910 11 -3 -5 -7 -9 -11 -2 -4 -6 -8 -10 1 2 3 4 5 6 7 Sally spends $ 10 for lemonade supplies. She charges $ 2 per cup. Number of cups Profit 0- 10 1- 8 2- 6 3- 4 4- 2 - 10 - 8- 6- 4- 2 y = 2 x – 10 NEXT = NOW + 2 ; STARTING AT - 10 x ≥ 0 y ≥ -10

16. SITUATION 10 (Waiting in Line) Xy TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENTPractical Domain Practical Range EQUATION 5 10 15 20 25 30 35 40 45 50 There are 10 people waiting in line before a store opens. Five people enter the line every two minutes. 5/2 10 Time (min) people 0 10 1 12.5 2 15 3 17.5 4 20 10 12.5 15 17.5 20 y = 2.5x + 10 NEXT = NOW + 2 ; STARTING AT 10 x ≥ 0 y ≥ 10

17. SITUATION 11 (Temperature (C o ) over time) Xy TABLE GRAPH SEQUENCE Initial Value 1st2nd3rd4th 73 SLOPE (rate of change) Y-intercept (initial value) NEXT-NOW STATEMENTPractical Domain Practical Range EQUATION 1 5 7 8 10 0 1 2 2 -2 3 3 -3 4 4 -4 5 -5 6 6 -6 7 -7 8 9 9 10 11 12 The temperature is 11 ° initially and drops 4 ° every hour. - 4 11 Time (hr) degrees 0 11 1 7 2 3 3 - 1 4 - 5 10 - 5 y = - 4 x + 11 NEXT = NOW – 4 ; STARTING AT 11 x ≥ 0 y ≤ 11