Math 8 Unit 5 Writing Equations Extra Help: Tuesday, November 20; 2:40pm Tuesday, November 27; 7:30am Tuesday, December 4; 7:30am Date Lesson Topic Homework M 11/19 1 Types of Slope Lesson 1 -Page 6 T 11/20 2 Finding slope algebraically Lesson 2 Page 10 W 11/21 3 Write an equation of a line (m and b) Lesson 3 Page 13 T 11/22 Happy Thanksgiving! F 11/23 No School! M 11/26 4 Write an equation of a line (graph) Lesson 4 Page 17 T 11/27 5 Functions Lesson 5 Page 24-25 W 11/28 6 Comparing Rate of Change Family Connect Night T 11/29 Quiz Review F 11/20 Quiz M 12/3 7 Graphing Systems of Equations Lesson 7 Page 35 T 12/4 8 Linear vs. Non Linear Lesson 8 Page 40-41 W 12/5 Review Check solutions & Study T 12/6 Test Name
Lesson 1 Types of Slope Aim: I can determine the various types of slope. Warm up: Rewrite in slope-intercept form and identify m and b. -15x + 3y = -3 m = b = Guided Practice: Slope rise run *Read from LEFT to RIGHT* Identify the type of slope and its value. type type type m = m = m = 2
type type type m = m = m = Draw a line using the given points and then identify the type of slope and its value. (*Be sure your points are labeled and your line has arrows!) 7) A (-5, 4) B (4, -3) 8) A (4, 3) B (4, -6) type type m = m = 3
Independent Practice: Draw a line using the given points and then identify the type of slope. 1) A (-3, 4) B (4, 4) 2) A (0, 3) B (4, -6) type m = type m = 3) A (-7, 4) B (4, 8) 4) A (3, 3) B (-3, -6) type m = type m = 4
Graph the line and then identify the type of slope. 5) y = 2x + 3 6) 3x + y = 7 Type Type 7) y = -4x + 9 8) y = 1 2 x - 5 Type Type 5
Lesson 1 Homework Identify the type slope. 1. 2. 3. 4. 5. Graph the line and then identify the type of slope. 6. y = 2x + 3 7. 3x + y = 7 Type Type 6
Aim: I can find the slope algebraically. Lesson 2 Finding Slope Algebraically Warm Up: Determine the rate of change (slope) for each graph. The of the line is the ratio of the vertical change of the line (change in y-values) to its horizontal change (change in x-values). This ratio is a constant rate of change between any two points on the line. Slope Formula: To calculate the slope of any line, all you need are two points that are on the line. Substitute the coordinates of the points into the formula slope (m) = y x (change of y) (change of y ) = the difference of the y coordinates the difference of the y coordinates = y 2 y 1 x 2 x 1 Note, the 1 and the 2 are not exponents, they are subscripts They correspond with the 1 st set of coordinates and the 2 nd set of coordinates Guided Practice: Determine the slope of the line given each set of points. (x1, y1) and (x2, y2) (x1, y1) and (x2, y2) 1. (2, 3) and (5, 6). 2. (-4, 8) and (6, 12) 7
(x1, y1) and (x2, y2) (x1, y1) and (x2, y2) 3. (6, 18) and ( 3, 18). 4. (-5, 8) and (6, 13) 5. (2, 2) and ( 4, 4). 6. (3, 5) and (-8, 5) Using the table below, determine the slope (rate of change) using the slope formula. Using the graph below, determine the slope using the slope formula. 7. x y 8. 1 3 2 6 3 9 4 12 Independent Practice: Determine the slope of the line given each set of points. 1. ( 6, 10) and (2, 10) 2. (1, 1) and (2, 4) 8
3. (2, 8) and (3, 11) 4. (0, 6) and (1, 4) 5. ( 2, 3) and (9,3) 6. (5,-2) and (5,1) Use table method: 7. (4,3) and (1,9) x y 8. (5,2) and (4,7) x Using the table below, determine the slope Using the graph below, determine (rate of change) using the slope formula. the slope using the slope formula. 9. x y 1 3 2 6 3 9 4 12 y 10. 9
Lesson 2 Homework Find the slope (rate of change) of the line containing the following points. 1) A(0,1) B(4,4) 2) A(4,7) B(8,8) 3) A(3,6) B(6,8) 4) A(4,1) B(4,9) 5) A(4,8) B(4,-1) 6) A(7,-1) B(-5,-4) Use table method: 7) A(0,-4) B(5,-1) x y 8) A(-1,-2) B(-1,-5) x y Using the tables below, determine the slope using the slope formula. 9) x y 10) x y 3 6 1 8-1 10-3 12 0 1 4 3 8 5 12 7 10
Lesson 3 Write the Equation of a Line Given m and b Aim: I can write an equation of a line given m and b Warm up: Find the slope and y-intercept of each: 1) y = 2x + 8 2) y = -4x + 3 3) y = -6x - 7 4) y = x + 2 m = m = m = m = b = b = b = b = Put into y = mx + b form (if needed). Then name the slope and y-intercept of each: 5) y - 4 = 2x 6) 3 + y = x 7) -3x + y = -4 8) -5 + 3y = 6x + 7 m = m = m = m = b = b = b = b = Guided Practice Write the equation of a line given the slope and y-intercept 1) m = 1 2) m = -2 3) m = 1 3 4) m = 0 b = -1 b = 3 b = 6 b = 4 5) slope = 3 6) slope = 4 7) slope = 2 3 8) slope = -4 y-int = -1 y-int = 4 y-int = -7 y-int = 0 11
Write the equation of a line given two points and the y-intercept 9) A(4,8) B(4,-1) 10) A(7,-1) B(-5,-4) m = m = b = -1 b = 3 Independent Practice: Write the equation of the line given slope (m) and y-intercept (b) 1) When m = 1 and b = -1 2) When m = -1 and b = 5 3) Having slope: 3; and y-intercept: -1 4) Having slope: 4; and y-intercept: 4 Write the equation of a line given two points and the y-intercept 5) A(-4,-1) B(-4,8) 6) A(4,-2) B(-2,1) 7) A(4,2) B(5,7) m = m = m = b = none b = 0 b = 2 12
Lesson 3 Homework Write the equation of the line given slope (m) and y-intercept (b) 1) Having slope: 3; and y-intercept: -1 2) Having slope: 4; and y-intercept: 4 3) Having slope: -2; and y-intercept: 6 4) Having slope: -4; and y-intercept: -4 5) When m = 1 and b = -1 6) When m = -1 and b = 5 7) When m = 2 and b = -5 3 8) When m = 3 and b = 0 9) Having slope: 3 and y-intercept: -5 5 10) When m = 0 and b = 5 11) Having slope: 1 4 and y-intercept: 2 12) When m = 5 7 and b = 0 Find the slope (m) then write an equation of the line. 13) A(2, -9) B(6, 3) 14) A (-3, 11) B(7,11) 15) A(0, 4) B(7, 3) m = m = m = b = -15 b = 11 b = none 13
Lesson 4 Write an Equation of a line from a Graph Aim: I can write an equation of a line from a graph Warm up: 1) The equation of a line is written as. 2) To write the equation of a line what 2 values do we need? & 3) The m represents the. 4) The b represents the. 5) The is the point where the line crosses the. Steps for Writing the Equation of a line from a graph 1 - Find the y-intercept (b) 2 - Find the slope (m) 3 - Write the equation of the line. Guided Practice: Write the equation of the line. 1) 2) 3) m = m = m = b = b = b = 14
4) 5) 6) Given points A(1,2) and B (-1,-2) Write the equation of the line. m = m = b = b = m = m = b = Independent Practice 15
Draw the line through the given points and then write the equation of the line. A (-3, 4) B (-3, -5) A( -2, 1) B (1, -5) A (-1, 3) B (1, -3) m = m = m = b = b = b = 16
Lesson 4 Homework Write the equation of the line. 4) Draw the line through the given points and then write the equation of the line. A (3, 3) B (-4, -4) A( -3, -3) B (1, 5) A (-4, -4) B (2, -1) 17
Aim: I can determine what is or is not a function. Lesson 5 - Functions Warm up: Draw the line through the points (2,4) and (6,6) and then write the equation of the line Vocabulary: Function: A set of ordered pairs in which each input value (x) has exactly one output value (y). You can check to see if a graph is a function by using the vertical line test. Vertical Line Test: If a vertical line is drawn and it intersects the graph once, it is a function. If the vertical line intersects the graph more than once, it is not a function. Guided Practice: Determine if the following are functions (function / not a function) 1) x y 2) x y 3) x y 4) 0-4 0 1 0 5 1-1 2 1 1 6 2 2 4 1 2 7 3 5 6 1 1 8 4 8 8 1 0 9 x y 12-2 10-1 8 0 10 1 6 2 18
5) x y 6) x y 7) x y 8) 5 4 3 4 3 6 3 8 3 2 4 7 2 5 2 3 5 8 8 0 6 Domain Sue Joe Emma Lilly Range Blue Red Pink Using the vertical line test state whether or not each graph is a function (yes / no) 9) 10) 11) 12) 13) Create a function rule (equation) to represent each situation 14) Emma is writing a term paper. She writes 3 pages per day. If y represents the total pages and x represent days, create a function rule for this scenario. 15) A large pool contains 20,000 gallons of water. 5 gallons evaporate each day. y is the gallons of water and x is the days. Write an equation. 16) In a bank account Joe has $45. He earns $2 per day. Write an equation for the situation. 19
Every Function can be written in four ways: Examples: 1. 2. 3. 4. 1) A) Write the equation B) Make a table from the line of the line x y C) Create a situation to represent this graph 2) x y A) Write the equation B) Graph the line 2-6 (function rule) of the line 4-2 6 2 8 6 C) Write a situation to model table. 20
3) y = 2x + 4 A) Make a Table B) Graph the line x y C) Write a situation to model table. 4) Amy has $10 in her piggy bank. She spends $1 per day on ice cream. A) Write an equation B) Make a Table C) Graph the line x y 21
Independent Practice Determine if the following are functions (function / not a function) 1) 2) 3) 4) x y Domain Range 5 2 A B 1 7 3 C 2 9 4 D 11 5 5 6 Which set of ordered pairs is not a function? 5) 6) 1) 2) 3) 4) Which set of ordered pairs represents a function? 1) 2) 3) 4) 7) A) Write the equation B) Make a table from the line of the line x y C) Create a situation to represent this graph 22
Create a function rule (equation) to represent each situation 8) Mr. Murphy has been working out. His bicep started out at 10. If he gains ½ per month, write an equation to match this situation. 9) You sell candy for a school fundraiser. You sell each box for $3. Write an equation for this scenario. 23
Lesson 5 Homework Determine if the following are functions (function / not a function) 1) 2) 3) 4) x y Domain Range Domain Range -2 2 Beth Dave 2 1-1 5 Sally Mike 3 2 Lucy Ryan 4 3 0 4 Jen Dan 5 1 5 x y 1 7 5 5 9 3 1 1 5) 6) 7) 8) 9) 3y = 3x a) Make a table from the equation b) Graph the equation x y c) Create a situation to represent this situation 24
Create a function rule (equation) to represent each situation 10) You open up a lemon-aide stand. You sell 13 lemon-aides per hour (x). Write an equation for this situation. 25
Lesson 6 Comparing Rate of Change (Slope) Aim: I can compare the rate of change of two situations. Warm up: Using the table below, determine the rate of change. x y 5 2 6 3 7 4 8 5 9 6 Vocabulary: Rate of Change: The measure of how quickly a linear equation increases or decreases. In a line, that is measured by determining the slope. NOTE: The steeper the slope, the GREATER the rate of change Determine the rate of change 1) y = -2x + 7 2) 3 x = y 3) 5y 7x = 45 4) 2y = 8x 6 5) 6) 7) 26
8) x y 9) x y 10) 5 11 0-1 6 9 2 7 7 7 4 15 8 5 6 23 9 3 8 31 x y 3 5 4 6 5 7 6 8 7 9 11) During a rain storm, it rained 0.5 per hour. 12) You buy a car with 3000 miles. Each year What is the rate of change? you drive an additional 11,500 miles per year. What is the rate of change? Guided Practice: Comparing Rates of Change 1) The following table and graph represent two different linear functions. Determine which has the greater rate of change. x y A) -4-3 OR B) -2-2 0-1 2 0 4 1 2) Which has the greater rate of change? A) y = x 6 OR B) 2y = 4x + 16 27
3) Which has the greater rate of change? A) A balloon deflates at 3 cubic OR B) centimeters per hour. At first the balloon is filled up to 30 cubic centimeters. hour deflate 0 30 2 25 4 20 6 15 4) In question #3, which example will have more air after 6 hours? Independent Practice: Determine the rate of change 1) 2y = -2x + 12 2) 3y 2x = 1 3) 4) 5) x y 6) 0 11 1 8 2 5 3 2 x y 2 5 4 8 6 11 8 14 m = m = 28
7) The hair on Bill s head grows 1.5 per month. His hair is 6 long. What is the rate of change? 8) The growth of the NY population increases by 95,000 per year. The current population is about 19.5 million. What is the rate of change? 9) The following table and graph represent two different linear functions. a. Write the equation of each b. Determine which has the greater rate of change x y A) OR B) -2 7 0 11 2 15 4 19 10) In question #9, which y will be the greatest when x = 3? 29
Lesson 6 Homework Determine the rate of change 1) 2) 3) 4) x y x y 5) 0 2-3 2 6) 1 5-1 4 5 17 0 5 12 38 9 14 x y 4 5 6 6 12 9 14 10 7) Compare the following functions. Write the equation of each function. Age of Child Weight of a Child (lb) 0 10 A) B) 2y = 10x + 28 C) When Joe was born he weighed exactly 8 lbs. 3 34 His doctor predicted he would gain 5.5 lbs per 4 42 year. 6 58 m = m = m = 8) Based on situations in #7, which of the three would have the greatest rate of change? 9) From question #7, at age, 10, which example will produce the child with the largest weight? 30
Lesson 7 Graphing Systems of Equations Aim: I can graph systems of equations (One Solution, No Solution, Infinitely Many Solutions) Warm up: 1) Graph and Solve y = -x + 2 y = 3x - 2 Solve for x: 2) 2x 4 = -x 1 3) 2x + 4 = 2(x + 1) 4) 2x + 4 = 2(x + 2) When you graph two lines, there are three things that can happen. The lines Intersect The lines Never Intersect The lines are the Same => One Solution => No Solution => Infinite Solutions 31
Guided Practice: Graph each and determine the type of solution. (one solution, no solution, or infinite solutions). 1) y = 2x + 4 2) y = x + 6 3) y = 2x + 1 y = 2x - 6 y = -2x 2y = 4x + 2 Type of Solution: Type of Solution: Type of Solution: How does it look graphically? How does it look graphically? How does it look graphically? What do you notice about the What do you notice about the What do you notice about the slopes and the y- intercepts? slopes and the y- intercepts? slopes and the y-intercepts? 32
Finding types of slopes algebraically (By Inspection) Use your knowledge of slopes and y-intercepts to determine the type of solution. (one solution, no solution, or infinite solutions). 4) y = 2x + 8 5) y = 3x + 8 6) y = 2x + 3 y = 2x 7 y = -2x 4 3y = 6x + 9 How do you know? How do you know? How do you know? Independent Practice: Solve graphically and state the type of solution: 1) y = 1 x 3 2) 3y 6x = 12 2 2y = x + 4 y = 2x + 4 Type of Solution Type of Solution 33
State the type of solution algebraically (by inspection) and explain how you got the answer. 3) y = x + 4 4) y = 2x + 3 5) y = 5x + 7 3y = 3x + 12 y = -2x + 9 y = 5x + 4 How do you know? How do you know? How do you know? 6) y = 3x + 2 7) 2y = 4x + 16 8) 5x 2y = 8 y = 3x + 6 y 2x = 8 5x 2y = -8 How do you know? How do you know? How do you know? Lesson Summary What are the characteristics of the graph that tell us how many solutions a system has? Slopes y-intercepts Lines One solution: No solution: Infinite solutions: 34
Lesson 7 Homework State the type of solution and then prove it graphically: 1) y = 4 x + 3 2) y = 2x + 4 3 y = 2 3 x 3 2y 8 = 4x Type of Solution Type of Solution State the type of solution algebraically and then explain how you got it. 3) 2y = 6x 10 4) y = 2x - 8 3y 9x = 12 y = 2x Type of Solution Explanation: Type of Solution Explanation: 35
Lesson 8 Linear vs. Non-Linear Aim: I can compare and contrast linear and non linear examples Warm up: Given the table to the right, determine the slope Linear Functions - A function in which the graph of the solution forms a straight line. A linear function has a constant rate of change. Non-linear Functions - The graph of a non-linear function is not a straight line. A non-linear function s rate of change in not constant. Remember: Every function can be written 4 ways: 1-2 - 3-4 - Guided Practice: State whether the following graphs are linear or non-linear 3) 4) 36
5) 6) 7) * 8) Are the following equations Linear or Non-linear 3 2 2 y x 3x 10) y x 5x 6 11) y 2x 10 12) y x x 2 9) 9 13) y = 5x 14) y = 2 15) y x 2 2x * 16) x = 8 Are the following tables Linear or Non-linear 17) 18) 19) 20) 37
Are the following word problems Linear or Non-linear (L / NL) 21) Sam put $10 in the box under his bed every week 22) A dolphin jumps above the surface of the ocean water, then dives back in the water. 23) A soccer player sprints from one side of the field to the other. 24) A lacrosse player throws a ball upward from her playing stick with an initial height of 7ft and an initial velocity of 90 ft. per second. 25) A rocket is shot off into the air and then comes back down to the ground. 26) Bill borrows $2,500 from the bank and has to pay it off monthly for 30 month Independent Practice: State whether the following equations Linear or Non-linear 1) y = x 2 x 2 2) y = x + 1 3) y = 5x + 2 4) y = x 3 3x + 9 5) y 7x = 2 38
Are the following tables Linear or Non-linear 6) 7) 8) 9) Are the following word problems Linear or Non-linear 10) A baseball player hits a pop fly 11) The path traveled by a basketball during a shot on the basket 12) A babysitter getting paid $6 per hour 13) You deposit $250 per year for 39 years Are the following graphs Linear or Non-linear 14) 15) 16) 17) 39
Are the following equations Linear or Non-linear Lesson 8 Homework 1) y = -27x + 2 15 2) y = 1 2 x2 + 5 3) y = x 4) y = 3 x Graph the equation for each function. Then complete the sentence 5) x y 6) -4-4 -2 0 0 4 2 8 4 12 The function is linear/ non-linear because The function is linear/ non-linear because 40
7) Which of the following does not describe a linear function? A. the perimeter, p, of a square with side s: p = 4s B. the circumference, C, of a circle with radius r: C = 2πr C. the salary, s, of an employee making $12.50 per hour, h: s = 12.50h D. the area, A, of a circle with radius r: A = πr 2 8) Which equation represents a linear function? A. y = 8x 4 B. y = 0.05x 0.01 C. y = 2x 2 + 5 3x D. x 41
Unit 5 Review 1) y = -6x - 4 2) y = x + 7 3) y = -3x 4) y = x + 10 slope = m = y-intercept = b = y-intercept = b = slope = m = Draw any line with the following slopes: 5) Positive Slope 6) Negative Slope 7) Zero Slope 8) Undefined Slope (no slope) Write the equation of a line when: 9) m = -9, b = 4 10) slope = 1/2, y-intercept = 0 11) m = 1, b = -3 12) slope = 4, y-intercept = 4 Solve for y and write the equation in y = mx + b form: 13) 4x + y = 12 14) -x + y = -6 15) 10 + 5y = 25x 42
16) Graph the line y = -3x + 5 slope = y - intercept = For numbers 17 & 18 graph the line that intersects points: 17) (-3,-2) and (2,-2) 18) (-4, 3) and (-4, -2) What type of slope is graphed? slope = y-intercept = What type of slope is graphed? slope = y-intercept = Determine the slope of the line containing the points: 19) (-5, 4) and (-2, 10) 20) (12, 15) and (8, 27) 21) (3, 6) and (-3, 6) 43
22) 23) 24) slope = slope = slope = y-intercept = y-intercept = y-intercept = equation of the line = equation of the line = equation of the line = Make a table for the above graph: Make a table for the above graph: Make a table for the above graph: x y x y x y Write the equation of the line: 25) x y 26) 4 10 10 22 16 34 22 46 28 58 x 4 5 6 7 8 y 5 7 9 11 13 rate of change = rate of change = 44
27) Place the following rate of changes in order from least to greatest. a) -3x + 6y = 18 b.) (2, 5) and (5,6) c.) d.) x y Taxis charge $2 per mile. 1 7 3 9 5 11 28) Determine if the following would be considered a function or NOT a function. a) b) c) d) x y x y 2 2 3 4 4 3 8 3 8 4 5 2 12 4 7 0 15 6 How do you know? How do you know? How do you know? How do you know? e) {(1, 2), (5, 7), (3, 4), (5, 2)} f) {(1, 3), (5, 3), (7, 4), (8,2)} How do you know? How do you know? 45
29) Determine if the following would be considered linear or non-linear. a) y = 5x + 1 b) -3x + y = 4 c) y = x 2 + 3x 10 d) x 2 + y 2 = 25 e) f) g) x 2 4 6 8 y 3 5 7 9 x 1 4 8 10 y 0 2 3 2 i) Mike makes $15 a week doing chores around the house. j) Jane served a volleyball over the net. 30) Draw 2 lines representing the following number of solutions: a) No Solutions b) One Solution c) Infinite Solutions 31) How many solutions does each system of equations represent? (one solution, no solution, or infinite solutions) a) y = -5x + 9 b) y = x + 5 c) y = 4x + 2 y = -5x 6 3y = 3x + 15 y = 3x 8 46
Unit 4 Review: 32) Graph the system of equations: y = x + 4 y = -2x + 7 33) What is the solution? 34) Check the solution: Unit 3 Review: Simplify. Rewrite using all positive exponents. 35) (2x 3 )(-5x) 36) (8 4 )(8-7 ) 37) 20x3 y 7 4x 5 y 2 38) (-2x4 y) 3 Unit 2 Review: Solve for x. State the type of solution (one, no solution, infinite). 39) 4(x 2) = 3x + 5 40) 0.75x + 1 = 0.25(3x + 4) 41) 5x - 3 2x = 4 + 3x 1 47
Unit 1 Review: 42) Simplify 10 2 x 5-2 3 43) Convert 59 F into Celsius using the formula C = 5 (F 32). 9 44) a) Find the area. b) Find the perimeter. 3x - 5 2x 48
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6. 7. 8. Slope = Slope = Slope = Type Type Type 1. 2. 3. Determine the type of slope for each line. 4. 5. 6. 57
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