Algebra 1 2 nd Six Weeks Second Six Weeks October 6 November 14, 2014 Monday Tuesday Wednesday Thursday Friday October 6 B Day 7 A Day 8 B Day 9 A Day 10 B Day Elaboration Day Test 1 - Cluster 2 Test Direct Variation -parent functions (constant/linear) -constant of variation 13 14 A Day 15 B Day 16 A Day 17 B Day Professional Development Day Slope -from a table, ordered pairs, equations, verbal PSAT Day 9-11 th Slope -from a table, ordered pairs, equations, verbal X and Y intercepts 20 A Day 21 B Day 22 A Day 23 B Day 24 A Day Applications Day (Changes to m & b) SKILL CHECK Graphing Linear Equations - Day 1 -use y = and f(x)= -from slope-intercept (solve for y and graph) -from standard form (x and y intercepts) Graphing Linear- Day 2 -transformations including function notation for specific a, b, c, and d values QUIZ 27 B Day 28 A Day 29 B Day 30 A Day 31 B Day Graphing Linear -Day 2 -transformations QUIZ Applications (Reasonable Domain and Range) Review Concepts -Slope -Intercepts -Graphing Linear -Direct Variation Test 2 Teacher Test Nov 3 A Day 4 B Day 5 A Day 6 B Day 7 A Day Writing Equations - Day 1 -given m and b -point/slope -given 2 points -from a table Writing Equations - Day 2 -verbal -go between forms Scatterplots -regression -correlation coefficient QUIZ 10 B Day 11 A Day 12 B Day 13 A Day 14 B Day Scatterplots -regression -correlation coefficient QUIZ Parallel & Perpendicular Lines Day 1 -use graphs to compare -write parallel and perpendicular lines -include horizontal and vertical -parallel and perpendicular with respect to the x & y axis (new) Parallel & Perpendicular Lines Day 2 1
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Variables x and y show when =. k is called the of. We say, varies with. Identify k 1 y = -5x y = x y = x 2 k = k = k = Sketch a graph for each equation. Create a table for each graph. What do you notice about each graph and each table? 3
Find the constant of variation. a. y = 3 when x = 24 b. y = 36 when x = -9 c. -2 50-1 25 3-75 4-100 6-150 Assume y varies directly with x. Find the specified value. a. If y = 8 when x = 4, find y when x = 12. b. If y = -15 when x = 5, find x when y = -4. Summary: 4
Name Date Direct Variation For each table find the constant of variation (slope) and corresponding equation. 1. x y 3 18 4 24 5 30 k = (slope) Equation 2. x 1 4 7 k = (slope) y 4 16 28 Equation Tell whether each illustrates a direct variation. If yes, why? 3. A person s arm span, and his or her height. 4. The number of eggs used in a particular recipe, and the number of cups of flour used. 5. The number of people sharing an apartment, and the number of dollars rent each one must pay a month. Find the constant of variation (k) for each. Assume y varies directly with x. 6. y = 12 when x = 24 7. y = 7 when x = 49 8. y = 36 when x = 4 5
Solve. Assume y varies directly with x. (Hint: Set up proportion) 9. If y = 2 when x = 6, find y when x = 24. 10. If x = 12 when y = 3, find x when y = 6. 11. If y = 7 when x = 2, find y when x = 8. 12. If y = 6 when x = 3, find x when y = 24. 13. Gary s wages vary directly with the times he works. If his wages for 5 hours are $60, what are his wages for 50 hours? 14. Kate earned $120 for 12 cars she washed. If the amount Kate earns varies directly with the number of cars she washed, how much will she earn for 5 cars? 6
Slope: Exploring Rate of Change Complete the table to show Heather s distance while bike riding. # of hours Calculation Total Distance Ordered Pair 2 12 3 18 4 24 5 30 1. What does the ordered pair (3, 18) mean? 2. What is the rate of change for Heather s bike ride? 3. Write the equation that describes the relationship between the number of hours (h) Heather rides her bike and the total distance she travels (d). 4. Graph the ordered pairs from the table above. Label each axis. 5. a. What is the change in miles from (2, 12) to (5, 30)? b. How did you get this answer? c. On the graph, is the change in miles vertical or horizontal? 7
6. a. What is the change in hours from (2, 12) to (5, 30)? b. How did you get this answer? c. On the graph, is the change in hours vertical or horizontal? miles 7. a. What is the speed as a fraction? hour b. How did you get this answer? c. How do the answers you got in 5a and 6a relate to speed? 8. How did we find this speed using our x and y values on the graph? (What is in the numerator and what is in the denominator?) 9. We are creating an equation we can use to find slope/rate of change anytime! What do you think the equation is? 10. Complete the chart and show how you can use the definition of slope to find the rate of change in the following table. x y 3 12.75 4 17 5 21.25 6 7 Slope = 8
H horizontal O zero slope Y crosses y-axis Y y = # V vertical U undefined slope X crosses x-axis X x = # y = - 4 x = - 3 I can find SLOPE (M) from Graph Table Ordered Pairs Equation x f(x) -1-2 2 4 4 8 5 10 8 16 (2, -3) (5, -6) y = -2x + 5 9
SLOPE (M) SLOPE (M) RATIO Division! SLOPE (M) SLOPE (M) Slope-Intercept Form y = Mx + B x f(x) -1-2 2 4 4 8 5 10 8 16 Ex: Find the slope: y = -2x + 5 Ex: Find the slope of the line that passes through the points (2, -3) (5, -6) y = ¼x + 3 y = x Rate of Change RATIO Steepness of a Line 10
Evaluate: Algebra I HW: Slope Name Date Period Plot the points and draw a line through them. Without calculating, tell whether the slope of the line is positive, negative, zero or undefined. 1. (1, -4) and (5, -8) 2. (-3, 6) and (-3, 0) a. Positive b. Negative c. Zero d. Undefined a. Positive b. Negative c. Zero d. Undefined 3. (7, 1) and (-2, 1) 4. (-4, -5) and (-3, -2) a. Positive b. Negative c. Zero d. Undefined a. Positive b. Negative c. Zero d. Undefined Find the slope value of the line that passes through the points. 5. Slope = 6. Slope = 7. Slope = 11
Find the slope value of the line that passes through the points represented in each table. 8. Slope = 9. Slope = 10. Slope = x y -5-3 1 2 7 7 x y 5-2 5 5 5 8 x y -1 2 4-1 9-4 Find the slope value of the line that passes through the points. 11. Slope = 12. Slope = 13. Slope = (3, 1) and (-5, 3) (3, 2) and (8, 2) (-5, -4) and (1, -2) Find the slope value from the following equations. 14.Slope = 15. Slope = 16. Slope = y = 2x + 7 y = -x -3 y = x + 8 Find the value of x or y so that the line passing through the two points has the given slope. 17. (-3, y), (-9, -2); m = 1 18. (x, -7), (1, 2); m = 3 19. (8, 1), (1, y); m = -1 20. (7, 5), (x, 2); m = 12
x- and y-intercepts What is an x-intercept? What is a y-intercept? A. What are the x- and y-intercepts of the graph below? Write you answer as an ordered pair. x-intercept y-intercept Find the x- and y-intercepts in ordered pair form for the following two graphs. x-intercept y-intercept x-intercept y-intercept B. Write the coordinates for the x- and y-intercepts for the following examples. 1. x-intercept is 4 2. y-intercept is 4 3. y-intercept is -1 4. x-intercept is -3.5 5. x-intercept is 0 6. y-intercept is -3/4 C. Graph the equation 2 x + 3y = 6 and then find the x- and y-intercepts. Use your calculator to find good points! x-int: y-int: x y 13
Given the table below, what is the x-intercept and the y-intercept? x- intercept: y intercept: x y 4 9 2 5 0 1 -.5 0 D. We can also find the intercepts using algebra rather than graphing. To find the x-intercept To find the y-intercept Find the x- and y-intercepts for the following examples using algebra. Show all work! 1. y = x 7 2 2. y = x + 6 3 3. 3 x 5y = 10 4. -3x +9y = 12 14
Algebra 1 Name Linear Equations Assignment Date Period Given Equation X-Intercept Y-Intercept graph Solve for Y Slope 1. -3x + y = 4 x-intercept (, ) y-intercept (, ) 2. x 2y = 4 x-intercept (, ) y-intercept (, ) 3. y = 4 x-intercept (, ) y-intercept (, ) 4. x = -2 x-intercept (, ) y-intercept (, ) 15
Given Equation X-Intercept Y-Intercept graph Solve for Y Slope 5. 2x y = -1 x-intercept (, ) y-intercept (, ) 6. -3x + y = 0 x-intercept (, ) y-intercept (, ) 7. x y = 3 x-intercept (, ) y-intercept (, ) 8. x + y = -1 x-intercept (, ) y-intercept (, ) 16
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Safari Hunt Treasure Map 18
Fill in the table for each situation. Interpreting Slope and Intercepts 19
Interpreting Slope and Intercepts 20
Interpreting Slope and Intercepts 21
Name Date Understanding Slope and Intercepts Identify the correct answer choice. Explain why the other answer choices are wrong. 1. The graph shows the amount of gasoline in a car s gas tank after x hours. Which statement is true? A. The car uses 2 gallons of gas each hour. B. The car uses 8 gallons of gas every 4 hours. C. The car has a 16 gallon tank. D. The gas tank is empty after 4 hours. 2. Alex s office has a water cooler. The table shows how many liters y of water are left in the cooler x days after it is filled. Which statement describes the x-intercept of the function that models the data? A. The cooler holds 36 liters of water. B. The cooler is empty after 3 days. C. People drink 9 liters per day. D. People drink 12 liters per day. 3. The graph of the line models the amount of memory remaining on Julia s mp3 player as she adds songs to the device. In this situation, what does the slope of the line represent? A. The memory remaining decreases at a rate of approximately 4.2 MB per song. B. The memory remaining increases at a rate of approximately 4.2 MB per song. C. The memory remaining decreases at a rate of approximately 42 MB per song. D. The memory remaining increases at a rate of approximately 42 MB per song. 22
4. In the equation y = 2x + 3, y represents the total number of preseason and regular season games that your softball team played this summer. The variable x represents the number of weeks in the regular season. Which statement is true? A. The regular season lasts 2 weeks. B. The preseason lasts 2 weeks. C. Your team plays 3 preseason games. D. Your team plays 3 regular season games each week. 5. A particular mosaic tile is being used in a kitchen backsplash design. The graph of the equation that models the situation shows the relationship between the number of tiles needed and the length of the design in feet. Which of the following best describes the meaning of the slope of the line for this situation? A. The length of the design increases at a rate of 12 tiles per foot. B. The length of the design increases at a rate of 10 tiles per foot. C. The length of the design increases at a rate of 6 tiles per foot. D. The length of the design increases at a rate of 2 tiles per foot. 6. The cost of a plumbing repair includes a $50 fee plus a cost of $15 per hour. In a graph of the equation to model this situation, what does the $50 fee represent? A. x-intercept B. y-intercept C. slope D. point of intersection 23
7. Belinda bought lemonade to serve at her birthday party. The graph models the relationship between the number of glasses of lemonade served and the amount of lemonade remaining. In this situation, what does the x-intercept represent? A. Belinda is serving 12 ounces of lemonade per glass. B. Belinda started with 320 ounces of lemonade. C. Belinda has enough lemonade for about 26 servings. D. Belinda can serve about 5 people per gallon. 8. The equation y = 4x + 5 gives the elevation y in meters of an elevator as a function of x seconds. Which statement describes the meaning of the slope of the function? A. The elevator starts at 4 meters above street level. B. The elevator starts at 5 meters above street level. C. The elevator rises 5 meters per second. D. The elevator rises 4 meters per second. 9. Sam is walking down the stairs in his office building. The equation y = -4x + 11 gives his elevation y in floors above the street after x minutes. What statement is true? A. Sam starts below street level. B. Sam descends 4 floors per minute. C. Sam starts 4 floors above street level. D. Same descends 11 floors per minute. 24
Name: Algebra 1 Graphing Lines Graphing Lines For questions #1 3, graph the line that contains the two points listed. 1. (-3, 1) and (2, 6) 2. (1,-2) and (- 1, 2) 3. (0, 4) and (0, 7) Graph the following lines, given the slope and a point on the line. 4. m = 2/5 ; (0, -1) 5. m = -3 ; (0, 5) 6. m = 1/4 ; (0, 0) 25
7. m = 0 ; (-1, 6) 8. m = -1 ; (6, -2) 9. m = undefined ; (5, 4) For #10-15, solve for y. Then graph the line. 2 10. y + 3 = -2(x 5) 11. y + 4 = (x 3) 12. 5x + 4y = 20 3 m= (, ) m= (, ) (, ) & (, ) 13. 3x + 4y = 12 14. X = 3 15. y = 5x + 3 y= Two Points: (, ) & (, ) y= 26
HW Graphing Equations: Name Date Period Solve the following problems for the variable y and graph them on the graph provided. 1) 5x = y + 4 2) y - 10 = -4x 3) 6x + y = 6 4) 4x + 3y = 12 27
5) 6x - 2y = 2 6) 9x - 3y = 3 7) y -4 = 2 (x + 3) 8) y = -4x + 6 28
Transformations to Linear Functions Engage 1. What is the linear parent function? 2. What is its slope? 3. What is its y-intercept? 4. Why is it considered a function? 5. What does its graph look like? 6. Where is its graph in relation to the origin? 7. What quadrants does its graph pass through? 8. What is its domain? 9. What is its range? 10. Why do you think it is called the parent function? 11. Compare and contrast the parent function to other lines that you have seen. 29
Changes to m and b EXPLORE 1. The parent function is y = x. Highlight (or circle) the part that is different on the y 2, and y 3 equations. 2. Sketch a graph of the following equations on the same coordinate plane, using a different color for each line. 1. y 1 = x y 2 = 2x y 3 = 5x 3. y 1 = x y 2 = x + 1 y 3 = x + 3 Describe the change of the slope and the change of the graph. Describe the change of the y-intercept and the change of the graph. 2. y 1 = x y 2 = 0.5x y 3 = 0.8x 4. y 1 = x y 2 = x 1 y 3 = x 5 Describe the change of the slope and the change of the graph. Describe the change of the y-intercept and the change of the graph. 30
Changes to and Meaning of m and b EXPLAIN Changes to m: What is the equation of the line on the graph? EQ: b = m = What is the difference between y = 6x + 1 and y = 2/3 x + 1? y = 6x + 1 m = b = What is the new equation if the slope is changed to 2? Graph this EQ. When the slope, the line got. y = 2 / 3 x + 1 m = b = When the slope, the line got. SUMMARY of changes to m: When the absolute value of the slope ( ) is increased, a line gets. When the absolute value of the slope ( ) is decreased, a line gets. When the sign of the slope ( ) is changed, the line changes. Changes to b: What is the equation of the line on the graph? EQ: b = m = What is the equation that results when you replace b with -3? What happens to the line? What kind of lines are these? What is the equation of the line on the graph? EQ: b = m = What is the equation of this line if it is translated 5 units up? What happens to the line? What kind of lines are these? SUMMARY of changes to b: When the y-intercept ( ) is changed, it results in a shift. A positive change means an shift. A negative change means a shift. 31
Meaning of m and b: 1. How do you find the y- intercept given the following linear representations a. an equation? 2. How do you find the slope given the following linear representations a. an equation? 3. What is the slope formula? b. a table or points? b. a table or points? 4. What does y mean? x c. a graph? c. a graph? Using this information, interpret the slope and y-intercept for each of the following: 5. a. What is the y-intercept? b. What does it mean? 6. a. What is the x-intercept? b. What does it mean? 7. a. What is the slope? b. What does it mean? 8. Label each axis as independent or dependent. 9. depends on. 10. How can the slope be interpreted given just the independent and dependent variables? Practice Equation: y = -25x + 500 11. What happens to the line if the slope changes to 50? 12. What happens to the line if the 500 is changed to a -40? 13. What would cause the line to shift up 15 units? 14. What would make the line flatter? 15. If the equation was a situation where amount of money in a savings account depends on the number of weeks, what is the meaning of the: slope? y-intercept? x-intercept? 32
Changes to and Meaning of m and b ELABORATE 1. A recording studio charges musicians a fee of $50 to record an album in their studio. Studio time costs an additional $35 per hour. Next year, the fee will go up to $100 for the recording and the hourly rate for studio time will change to $25. Original EQ: New EQ: What effects would these changes have on the graph? Meaning of the slope: Meaning of the y-intercept: 2. # bottles recycled 9 33 57 96 $ made 0.45 1.65 2.85 4.80 Slope and meaning: y-intercept and meaning: What would happen to the graph if you earned $0.07 per bottle you recycled? 3. Irma has a 12-gallon bucket. She fills the bucket with 9 gallons of water. The bucket has a crack and leaks water. x-intercept with meaning: y-intercept with meaning: Slope with meaning: Equation: Independent: Dependent: Domain: Range: What would happen to the graph if she only filled the bucket with 6 gallons? Andrew sells electronics. He earns 14% commission on his daily sales. x-intercept with meaning: y-intercept with meaning: Slope with meaning: Equation: Independent: Dependent: Domain: Range: What would happen to the graph if he earned 15% commission? 33
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Changes to and Meaning of m and b EVALUATE Name: For #1-3, consider the following graph 1. What are the coordinates of the x-intercept? 2. If the x-intercept stays the same and the y-intercept is changed to (0, -6), What happens to the slope of the line? 3. If the y-intercept stays as shown and the x-intercept becomes (-8, 0), what happens to the slope of the line? Give the new Equation for each line. 4. y = 5x + 9 is translated 7 units down 5. y = 1 x is translated 5 up. 6. y = -8 + 3 / 2 x is translated 2 units up 7. y = x is translated 8 units down For #8-12, a submarine was at -195 meters and then it rose at 12 meters per minute. 8. Write the equation of the line. Find the reasonable domain and range. EQ: Reasonable Domain: Reasonable Range: 9. How would the slope of the line change if the -195 were changed to -250. a. The line would be steeper b. The line would be less steep c. The line would have the same slope 10. How would the slope of the line change if the 12 were changed to 10. a. The line would be steeper b. The line would be less steep c. The line would have the same slope 11. How would the y-intercept of the line change if the -195 were changed to -180. a. It would translate up 15 b. It would translate down 15 c. It would stay the same 12. How would the y-intercept of the line change if the 12 were changed to 15. a. It would translate up 3 b. It would translate up 15 c. It would stay the same 35
Dan visits his grandmother in Alaska. On Sunday he hears on the radio that the temperature in his grandmother s city way -2 F from 8:00 AM until 4:00 PM. x-intercept with meaning: y-intercept with meaning: Slope with meaning: Equation: Independent: Dependent: Domain: Range: What would happen to the slope if the temperature was -7 F throughout the day? A beetle is 28 inches underground. The beetle digs straight upwards, in a vertical line, at a constant rate until it reaches the surface. x-intercept with meaning: y-intercept with meaning: Slope with meaning: Equation: Independent: Dependent: Domain: Range: What would happen to the x- intercept if the slope was changed to 3? The popcorn kernels weigh 30 ounces. Betsy pours the kernels into her popcorn maker at a constant rate. x-intercept with meaning: y-intercept with meaning: Slope with meaning: Equation: Independent: Dependent: Domain: Range: What would change about the graph if the slope was -3? Dawn buys plastic beads in bulk. The beads sell for $3 a pound, including sales tax. x-intercept with meaning: y-intercept with meaning: Slope with meaning: Equation: Independent: Dependent: Domain: Range: What would change about the graph if there was a shipping fee of $5? 36
Finding Intercepts with Meaning Name: Equation x-intercept with y-intercept with meaning meaning A bird is 32 meters from its next. It flies further away at a rate of 1 meter per second. Slope with meaning Reasonable Domain: Reasonable Range: Equation A beetle is 28 inches underground. The beetle digs straight upwards, in a vertical line, at a constant rate of 4 inches per minute until it reaches the surface. x-intercept with meaning y-intercept with meaning Slope with meaning Reasonable Domain: Reasonable Range: Equation Andrew sells electronics. He earns 14% commission on his daily sales. x-intercept with meaning y-intercept with meaning Slope with meaning Reasonable Domain: Reasonable Range: 37
Equation Irma has a 12-gallon bucket. She fills the bucket with 9 gallons of water. The bucket has a crack. The graph shows the remaining amount of water with respect to time. x-intercept with meaning y-intercept with meaning Slope with meaning Reasonable Domain: Reasonable Range: Equation Dan visits his grandmother in Alaska. On Sunday he hears on the radio that the temperature in his grandmother s city way -2 degrees F from 8:00 AM until 3:00 PM. x-intercept with meaning y-intercept with meaning Slope with meaning Reasonable Domain: Reasonable Range: You spent $500 setting up a jewelry business and charge $20 per necklace. Define Variables: x = y = Equation: x-intercept with meaning y-intercept with meaning Slope with meaning Reasonable Domain: Reasonable Range: A hot air balloon is at 200 feet and is descending at 5 feet per minute. Define Variables: x = y = Equation: x-intercept with meaning y-intercept with meaning Slope with meaning Reasonable Domain: Reasonable Range: 38
Name Date Changes to M and B Identify the correct answer choice. 1. The graph of y = x + 1 is shown. Which point will lie on the graph if the slope of the line is doubled and the y-intercept stays the same? A. (-2, -2) B. (0, 2) C. (1, 2) D. (2, 5) 2. Suppose the y-intercept of y = 2x 3 doubles but the slope remains the same. Which point will lie on the graph of the new line? A. (1, 1) B. (2, 2) C. (1, -2) D. (2, -2) 3. The graph of y = 4 1 x 2 is shown. Which point will lie on the graph if the slope of the line is 5 5 tripled and the y-intercept is increased by 1? A. (-2, 3 3 ) B. (0, 5 4. Which line has the steepest slope? 1 3 ) C. (1, 3) D. (2,-4) 5 A. y = 3x + 5 B. y = x 8 C. y = -x + 10 D. y = -6x 4 39
5. What happens to the graph of y = -3x + 2 when the y-intercept is decreased by 2? A. The new line is half as steep. B. The graph rises from left to right. C. The graph is 2 units higher for each value of x. D. The graph passes through the origin. 6. What happens to the graph of y = x 3 when the slope is divided by 2? A. The new line is half as steep. B. The new line is twice as steep. C. The graph is 2 units higher for each value of x. D. The graph is 2 units lower for each value of x. 7. The graph of a linear function is shown below. Another linear function, with a graph steeper than the one shown, is to be added to the same grid. Which equation could represent the new function? A. y = 3 5 x B. y = 5 3 x C. y = 1.25x D. y = 2.75x 40
8. How does multiplying the slope by -1 change the graph of y = -x + 1? A. The line then rises from left to right. B. The line then falls from left to right. C. The y-intercept becomes -1. D. The y-intercept is 1 unit less for every value of x. 9. The original function y = 3 2 x 7 is graphed on the same grid as the new function y = 4 3 x 7. Which of the following statements about these graphs is true? A. The graph of the original function is steeper than the graph of the new function. B. The graph of the original function is parallel than the graph of the new function. C. The graph of the new function is steeper than the graph of the original function. D. The graph of the new function is the same as the graph of the original function. 10. How does the graph of y = 0.75x 4 change when the y-intercept is increased by 0.25? A. The slope becomes 1. B. The line becomes less steep. C. The graph is 0.25 unit higher for every value of x. D. The graph is 0.25 unit lower for every value of x. 11. If the slope of the equation y = 3 x 2 is 4 changed to y = 4 3 x and the y-intercept remains the same, which statement best describes this situation? A. The new line is perpendicular to the original line. B. The new line is parallel to the original line. C. The new line and the original line have the same graph. D. The new line and the original line will intersect at (0, -2). 41
12. The function y = 6x 4 is graph below. If the slope of the function is multiplied by 0.5, and the y-intercept remains the same. Which of the following graphs best represents the new function? A. B. C. D. 42