MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

Transcription

1 FOM 11 T6 RATES AS GRAPHS 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) GRAPH a visual representation of a relationship between two different quantities. 2) SLOPE m a measure of the steepness of a graph or a part of a graph. 3) Δ a capital Greek letter delta and in mathematics and science it means the change in. i.e. Δy means the change in y 4) RATE a fraction composed of two quantities. e.g. 100 km $ g 55 words lbs 1 kg 5) UNIT RATE a special rate where the number of one of the quantities is a 1. e.g. 100 km 6) CONVERSION FRACTION a fraction used to convert one unit within a rate to another unit. e.g. 1 km 1000 m 60 s 1 1 kg 2.2 lb 55 words 1 1 kg 2.2 lbs I) REVIEW OF SLOPE A) Study the graph given in EXAMPLE 2 on page 454 of your text. The graph describes a change in distance over a change in time. The change in time is given on the horizontal axis (x) while the change in distance is given on the vertical axis (y). The graph is composed of 4 different sections each beginning and ending at specific times and distances. Each line-section has a unique steepness. Mathematicians detere the steepness of a straight-line graph by calculating its SLOPE. 1) The slope of a straight-line graph is calculated using this formula: slope m Δy Δx y y 2 1 change in the veritcal variable x 1 change in the horizontal variable ; slope is often stated as rise run a) In order to calculate the slope of a straight-line graph or a part of it you must know two points that are on the line. The points are always written in the form P 1 (x 1 y 1 ) and P 2 ( y 2 ). The coordinates of these points are substituted into the formula to calculate slope. B) SAMPLE PROBLEM 1: Study this example carefully. Be sure you understand and memorize the process used to complete it. 1) List two different sets of points that can be used to calculate the slope of the graph found in the top grid on page 296 of your text then calculate the slope using each set. What conclusions can you make when you compare the two slopes? Step 1: List two points on the line as P 1 and P 2 : A) P 1 ( 5 0) and P 2 (0 2) B) P 1 (0 2) and P 2 (5 4) x 1 5 y 1 0 and 0 y 2 2 x 1 0 y 1 2 and 5 y 2 4 Step 2: Calculate the slope using P 1 and P 2 : slope A y 2 y slope B y 2 y Step 3: Answer the question: The slopes are the same. B) REQUIRED PRACTICE 1: Detere the slope of the line found in the graphs indicated below. All page numbers are from your text. {Answers are on page 4 of these notes.} 1) Most left-hand graph on page ) Graph found in Question 4c on page ) Horizontal line found in the grid on page ) Dashed graph found in the bottom grid on page ) Solid line found in the middle grid on page ) Dashed graph in Question 1 on page 347.

2 FOM 11 T6 RATES AS GRAPHS 2 I) DETERMINING A RATE FROM A GRAPH A) Study the graph titled Distance vs. Time on page 454. NOTICE that each axis has represents to a different quantity. The horizontal axis represents Time having units in utes () while the vertical axis represents Distance having units in kilometres (km). Including the units in the slope calculation turns the slope into a rate and can be used to learn information about the situation described by the graph. B) SAMPLE PROBLEMS 2: Study this example carefully. Be sure you understand and memorize the process used to complete it. 1) Turn your text to page 454 and use the graph given in EXAMPLE 3 to answer these questions. a) Detere the rate speed in km/ and km/h for the first section of the graph: between 0 and 30. P 1 (0 0) and P 2 (30 2) x 1 0 y 1 0 and 30 y 2 2 slope y 2 y 1 ( 2 0 ) km ( 30 0 ) 15 2 km km! km 60! 8 km The rate speed of the first section of the graph is or 8 km. b) Detere the rate speed in km/ and km/h for the section of the second graph: between 30 and 40. P 1 (30 2) and P 2 (40 2) x 1 30 y 1 2 and 40 y 2 2 slope y 2 y 1 ( 2 2 ) km ( 40 30) 10 0 km 0 km 0 km 60! 0 km The rate speed of the second section of the graph is 0 km or 0 km. c) Detere the rate speed in km/ and km/h for the third section of the graph: between 40 and 60. P 1 (40 2) and P 2 (60 5) x 1 40 y 1 2 and 60 y 2 5 slope y 2 y 1 ( 5 2 ) km ( 60 40) 20 3 km! 60! 9 km The rate speed of the first section of the graph is or 9 km.

3 FOM 11 T6 RATES AS GRAPHS 3 d) Detere the rate speed in km/ and km/h for the fourth section of the graph: between 60 and 65. P 1 (60 5) and P 2 (65 0) x 1 60 y 1 5 and 65 y 2 0 slope y 2 y 1 ( 0 5 ) km ( 65 60) 5 km 5 1 km 1 km 60! 60 km Since speeds are always recorded as a positive number the rate speed of the fourth section of the graph is 1 km or 60 km. h 2) What is the relationship between the steepness of the graph and the numerical value of the slope? The steepest part of the graph section 4 has the largest slope number of 1 km larger rate. The smaller slope of slope rate: section 1. When the slope is zero 0 km or 8 km or 60 km h thus a corresponds to the section of the graph having a lower or 0 km the graph is flat. THE STEEPER THE GRAPH THE LARGER THE SLOPE AND THUS THE LARGER THE RATE. THE LESS STEEP THE GRAPH THE LOWER THE SLOPE AND THUS THE LOWER THE RATE. 3) Turn your text to page 451 and use the graph titled Distance vs. Time to answer these questions. a) Which graph represents the faster speed (rate)? Explain your answer. The graph describing the journey from Cold Lake to Reserve #149 the blue graph has the faster speed rate because its graph is steeper than the graph describing the journey from Reserve #149 to Bonnyville the red graph. b) Which graph represents the slower speed (rate)? Explain your answer. The graph describing the journey from Reserve #149 to Bonnyville the red graph has the slower speed rate because its graph is not as steep as the graph describing the journey from Cold Lake to Reserve #149 the blue graph. 4) Turn your text to page 457 and use the graph titled Gas Used vs. Distance Driven for Mario s trip to answer these questions. a) Which graph represents the higher rate? Explain your answer. The graph describing leg 2 the red graph has the higher rate because its graph is steeper than the graph describing leg 1 the blue graph. b) Which graph represents the slower lower rate? Explain your answer. The graph describing leg 1 the blue graph has the lower rate because its graph is steeper than the graph describing leg 1 the red graph.

4 FOM 11 T6 RATES AS GRAPHS 4 B) REQUIRED PRACTICE 2 1) Pages : Questions & 14. {Answers are on page 580 of the text.} 2) Page 473: Questions 5. {Answers are on page 581 of the text.} ANSWERS TO THE REQUIRED PRACTICE Required Practice 1 from pages 1 1) m 2 2) m 1 3) m 0 4) m 3 2 5) m 1 2 6) m 5 2

5 FOM 11 T6 RATES AS GRAPHS 5 LAST then FIRST Name T6 RATES AS GRAPHS Block: Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.) REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! Answer these questions. BE SURE TO SHOW ALL CALCULATIONS!!` 1) The graph below represents the analysis of a cyclist by their trainer. Use it to answer the questions below. Use it to answer the questions below. Record all answers to two decimal places. BE SURE YOU JUSTIFY YOUR ANSWERS WITH THE APPROPRIATE CALCULATIONS AND NUMBERS. a) Calculate the rate the cyclist travelled over each interval (All six). (12) b) Over which interval is the cyclist travelling the slowest. Explain your choice. (2) c) Over which interval is the cyclist travelling the fastest. Explain your choice. (2) d) Does the cyclist travel the same speed over during more than one interval? Explain your choice. (2) e) Is the cyclist travelling faster during interval EF or FG? Explain your choice. (2) f) What is the total distance the cyclist travelled? Explain your choice. (2) g) What is the total time the cyclist travelled? (1) 2) A container is being filled using water from the tap. The water is running at a constant rate. The graph below shows the depth of the water in the container over time. Which container is being filled? (1) /24