Chapter 5 Linear Functions Describing Lines Using Algebra

Section 5 - 1 Rate of change & Slope Objectives: To find rates of change from tables To find slope

WARM UP

Rate of Change: Shows the relationship between two changing quantities. When one quantity depends on the other, the following is true: Rate of Change = 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒕𝒉𝒆 𝑫𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕 𝑽𝒂𝒓𝒊𝒂𝒃𝒍𝒆 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒕𝒉𝒆 𝑰𝒏𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕 𝑽𝒂𝒓𝒊𝒂𝒃𝒍𝒆

Problem # 1: Finding Rate of Change Using a Table A) The table shows the distance a band marches over time. Is the rate of change in distance with respect to time constant? What does the rate of change represent?

Problem # 1: Finding Rate of Change Using a Table B) Do you get the same rate of change if you use nonconsecutive rows of the table? Explain.

Problem # 1: Finding Rate of Change Using a Table C) The table shows the elevation of a hang glider over time. Is the rate of change in elevation with respect to time constant? What does the rate of change represent?

The graphs of the ordered pairs (time, distance) in Problem 1 lie on a line, as shown below. The relationship between time and distance is linear. When data are linear, the rate of change is constant.

Notice that the rate of change in problem 1 is just the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The rate of change is called the slope of the line. Rate of Change: 𝟐𝟔𝟎 𝒇𝒆𝒆𝒕 𝟏 𝒎𝒊𝒏𝒖𝒕𝒆

Slope = 𝑽𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝑪𝒉𝒂𝒏𝒈𝒆 𝑯𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝑪𝒉𝒂𝒏𝒈𝒆 = 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏

Problem # 2: Finding Slope Using a Graph What is the slope of each line? A) B)

Problem # 2 Got It? What is the slope of each line? 1) 2) 1) 2) Pick a different point on the line to find the slope for # 2. Do you get the same slope?

HOMEWORK Textbook Page 298 – 299; #8 – 16, 30 - 32

8) No 9) yes, 1, 1 bun per hot dog 10) yes, -500, the plane is descending at a rate of 500 feet per minute 11) -2 14) 𝟓 𝟔 12) 𝟏 𝟑 15) 𝟑 𝟒 13) 4 16) − 𝟓 𝟐 30) Independent variable: Time (hours) Dependent variable: Snow (meters) Rate of change: .02 meters per hour 31) Independent variable: # of people Dependent variable: Cost Rate of change: $12 per person 32) Independent variable: time (hours) Dependent variable: distance (km) Rate of change: -60 km per hour (getting closer to destination)

Objectives: To find slope Section 5 - 1 Continued Objectives: To find slope

SLOPE You can use any two points on a line to find slope. Use subscripts to distinguish between the two points. Slope Formula: The x-coordinate you use first in the denominator must belong to the same ordered pair as the they y-coordinate you use first in the numerator

Problem # 3: Finding Slope Using Points A) What is the slope of the line through (-1, 0) & (3, -2)?

Problem # 3: Finding Slope Using Points What is the slope of the line through (1, 3) & (4, -1)? C) Plot the points from (B) and draw a line through them. Does the slope of the line look as you expected it to? Explain.

What is the slope of the line through F(-3, 2) & G(1, 5)? Problem # 3 Got It? What is the slope of the line through F(-3, 2) & G(1, 5)?

Problem # 4: Finding Slopes of Horizontal & Vertical Lines What is the slope of each line? A) B)

Problem # 4 Got It? What is the slope of the line through the given points? (4, -3) (4, 2) (-1, -3) (5, -3)

Problem # 5: Determining the Slope of a Line for Real-Life Situations Without graphing, tell whether the slope of a line that models each linear relationship is positive, negative, zero or undefined. Then find the slope. A babysitter earns $9 for 1 hour and $36 for 4 hours. The length of the trip to LBI is 60 miles long the 1st time and 60 miles long the 4th time.

Problem # 5 Got It? Without graphing, tell whether the slope of a line that models each linear relationship is positive, negative, zero or undefined. Then name the slope. Robi has run the first 4 miles of a race in 30 minutes. She reached the 6 mile point after 45 minutes.

Problem # 6: Finding a Missing Coordinate Each pair of points lines on a line with the given slope. Find x or y. A) (3, y) (1, 9); slope = − 𝟓 𝟐

Problem # 6: Finding a Missing Coordinate Each pair of points lines on a line with the given slope. Find x or y. B) (3, 5) (x, 2); Slope Undefined

Problem # 4 Got It? Each pair of points lines on a line with the given slope. Find x or y. (4, 3) (5, y); slope = 3

Ticket Out State the independent and dependent variable. Then find the rate of change. The distance of a cyclist is 1120 feet after 1 minute and 3360 feet after 3 minutes. What is the slope of the line shown? Find the slope of the line that goes through (4, 5) and (6, 2) Name and draw an example of each type of slope.

HOMEWORK Textbook Page 297 – 299; #1-7 all, 17 – 25 all, 26, 28, 42 – 46 Even

Yes, the rate of change between any two points is the same -1/5 -5/3 Slope; slope is the ratio of vertical change to horizontal change Speed staying constant for a period of time; 0; the slope of a horizontal line is 0 Need the graph to count, need the coordinates to use the formula. Either way you will get the same slope They did horizontal change over the vertical change.

1 42) 0 ½ 44) -6 -1 46) 12 2 7/10 -1/3 Undefined Zero 28) Negative; -2

WARM UP An airplane descended 2000 feet in 20 minutes and 3500 feet in 5 minutes. Name the dependent and independent variables. Then find the rate of change. The cost of tickets to the movies is $12 for 1 ticket and $48 for 4 tickets. Without graphing, tell whether the slope of a line that models this situation is positive, negative, zero or undefined. The points (7, 4) and (3, y) lie on a line that has a slope of 𝟏 𝟒 . Find y.