Pg #3, 4-24eoe, 36-39, 46, 51-52

Quiz 1 Pg. 232 #4-12e, 13

Slope of a Line Rate of Change – In the Real WorldIn the Real World Slope - RapRap Video of importance of slope and what we will be learning: ow/52023http://store.discoveryeducation.com/product/sh ow/52023

x y Two points on a line are all that is needed to find its slope. slope = = = F INDING THE S LOPE OF A L INE The slanted line at the right rises 3 units for each 2 units of horizontal change from left to right. So, the slope m of the line is The slope m of a nonvertical line is the number of units the line rises or falls for each unit of horizontal change from left to right. nonvertical line (2, 5) rise = = 3 units rise run run = = 2 units (0, 2)

x y run change in x The slope m of the nonvertical line passing through the points (x 1, y 1 ) and (x 2, y 2 ) is Read y 1 as “y sub one” Read x 1 as “x sub one” m = = (x1, y1)(x1, y1) (x 1, y 1 ) (x 2, y 2 ) (x2, y2)(x2, y2) y2 - y1y2 - y1 (y 2 - y 1 ) x2 - x1x2 - x1 (x 2 - x 1 ) F INDING THE S LOPE OF A L INE (x2, y2)(x2, y2) (x1, y1)(x1, y1) rise change in y =

m = run change in x rise change in y = = y2 - y1y2 - y1 x2 - x1x2 - x1 y2 - y1y2 - y1 m = x2 - x1x2 - x1 The order of subtraction is important. You can label either point as (x 1, y 1 ) and the other point as (x 2, y 2 ). However, both the numerator and denominator must use the same order. numerator y2 - y1y2 - y1 denominator x2 - x1x2 - x1 F INDING THE S LOPE OF A L INE When you use the formula for the slope, Subtraction order is the same the numerator and denominator must use the same subtraction order. CORRECT x1 - x2x1 - x2 y2 - y1y2 - y1 Subtraction order is different INCORRECT

Exploring Slope Animated Activity: 7_na/resources/applications/animations/explore_lea rning/chapter_4/dswmedia/4_4_RiseRun.html

Slopes for Horizontal and Vertical Lines Horizontal Lines have equation: y = some #, so slope is 0 –Ex: y = 3 two points are on line: (0, 3) (1, 3) Vertical Lines have equation: x = some #, so slope is UNDEFINED –Ex: x=2 two points are on line: (2,0) (2,1)

Determining Slope By Looking at a Graph Lines that go UP left to right = POSITIVE slope Lines that go DOWN from left to right = NEGATIVE slope Horizontal lines = ZERO slope Vertical lines = UNDEFINED slope

Slope and Related Equation

EXAMPLE 1 Find a positive slope Let (x 1, y 1 ) = (–4, 2) = (x 2, y 2 ) = (2, 6). m = y 2 – y 1 x 2 – x 1 6 – 2 2 – (–4) = = = Simplify. Substitute. Write formula for slope. Find the slope of the line shown.

EXAMPLE 2 Write an equation from a graph GUIDED PRACTICE for Example 1 Find the slope of the line that passes through the points. ANSWER 3 1. (5, 2) and (4, –1) 2. (–2, 3) and (4, 6) 1 2 ANSWER 3. (, 5) and (, –3) ANSWER 2

EXAMPLE 2 Find a negative slope Find the slope of the line shown. m = y 2 – y 1 x 2 – x 1 Let (x 1, y 1 ) = (3, 5) and (x 2, y 2 ) = (6, –1). –1 – 5 6 – 3 = – 6 3 = = –2–2 Write formula for slope. Substitute. Simplify.

EXAMPLE 3 Find the slope of a horizontal line Find the slope of the line shown. Let (x 1, y 1 ) = (–2, 4) and (x 2, y 2 ) = (4, 4). m = y 2 – y 1 x 2 – x 1 4 – 4 4 – (–2) = 0 6 = = 0 Write formula for slope. Substitute. Simplify.

EXAMPLE 4 Find the slope of a vertical line Find the slope of the line shown. Let (x 1, y 1 ) = (3, 5) and (x 2, y 2 ) = (3, 1). m = y 2 – y 1 x 2 – x 1 Write formula for slope. 1 – 5 3 – 3 = Substitute. Division by zero is undefined. ANSWER Because division by zero is undefined, the slope of a vertical line is undefined. – 4 0 =

EXAMPLE 2 Write an equation from a graph GUIDED PRACTICE for Examples 2, 3 and 4 Find the slope of the line that passes through the points. 4. (5, 2) and (5, –2) ANSWERundefined 5. (0, 4) and (–3, 4) 0 ANSWER 6. (0, 6) and (5, –4) ANSWER –2–2

EXAMPLE 5 Find a rate of change INTERNET CAFE The table shows the cost of using a computer at an Internet cafe for a given amount of time. Find the rate of change in cost with respect to time. Time ( hours ) 246 Cost ( dollars ) Rate of change = change in cost change in time 14 – 7 4 – 2 = 7 2 = 3.5 = ANSWER The rate of change in cost is $3.50 per hour. SOLUTION

Time ( minute ) Distance ( miles ) GUIDED PRACTICE for Example 5 The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time. 7. EXERCISE ANSWER 0.05 mi / min

EXAMPLE 6 Use a graph to find and compare rates of change COMMUNITY THEATER A community theater performed a play each Saturday evening for 10 consecutive weeks. The graph shows the attendance for the performances in weeks 1, 4, 6, and 10. Describe the rates of change in attendance with respect to time.

SOLUTION EXAMPLE 6 Use a graph to find and compare rates of change Find the rates of change using the slope formula. Weeks 1–4: 232 – – 1 = = 36 people per week Weeks 4–6: 204 – – 4 = –28 2 = –14 people per week Weeks 6–10: 72 – – 6 = –132 4 = –33 people per week ANSWER Attendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly.

EXAMPLE 7 Interpret a graph COMMUTING TO SCHOOL A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words. The first segment of the graph is not very steep, so the student is not traveling very far with respect to time. The student must be walking. The second segment has a zero slope, so the student must not be moving. He or she is waiting for the bus. The last segment is steep, so the student is traveling far with respect to time. The student must be riding the bus. SOLUTION

EXAMPLE 7 Interpret a graph GUIDED PRACTICE for Examples 6 and 7 WHAT IF? How would the answer to Example 6 change if you knew that attendance was 70 people in week 12? 8. Sample answer: The attendance did not decrease as rapidly between weeks 10 and 12. ANSWER WHAT IF? Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. 9. ANSWER

NAME

Check Yourself Pg #4-18 e, 22-28e, 36-42e, 43