Unit Calendar. Date Sect. Topic Homework HW On-Time April Direct Variation Page 256: #3-37 odd

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1 Name/Period: Unit Calendar Date Sect. Topic Homework HW On-Time April Direct Variation Page 256: #3-37 odd Essential Question: What does the constant of variation represent in direct variation? April Inverse Variation Page 769: 1-47 odd Essential Question: What does the constant of variation represent in inverse variation? April 25 Unit 10 Review Study for Quiz Essential Question: How do you represent direct/inverse variation algebraically? April 27 Unit 10 Quiz Essential Question: How do you determine whether inverse/direct variation exists? Page 1 of 12

2 Page 2 of 12

3 Graphing Direct Variation Equations Direct Variation equations are in the form y = kx. "Y varies directly with x" or "x and y vary directly" mean the same thing. is the input value. is the coefficient of x and is called the constant of variation. is the output value. The constant of variation (k) is also the slope of the graph of the equation! Reminder, the slope of a graph is the ratio of: change of y change of x To measure the slope we measure the distance it goes up or down, over the distance it goes right or left. We ll use our knowledge of slope to quickly and easily graph direct variation equations. Step 1: If necessary, rewrite the equation in y = kx form. Step 2: If necessary, rewrite the slope as a fraction. Ex: 2 becomes becomes ½ becomes 3 2 Step 3: Find the y-intercept and graph the point. Step 4: Use the slope ratio to find at least one other point to graph. Remember to always move up or down first, then left or right. Page 3 of 12

4 Graph each equation using the above method. 1) y = 2x 2) y = -4x 3) y = ¾x 4) y = -⅓x Page 4 of 12

5 5) y = - 3x 6) y = 4 5x Page 5 of 12

6 Writing Direct Variation Equations Reminder: direct variation equations are in the form y = kx. We say that y varies directly with x. If you know one set of coordinates you can write the equation that is true for all! Step 1: Write the equation y = kx. Step 2: Substitute the given values for x and y. Step 3: Solve for k. Step 4: Go back to y = kx and substitute the value found in step 3 for k. Examples: Write an equation for each. Y varies directly with x. a) When x = 2, y = 8 b) When x = 10, y = 5 1) y = kx 1) 2) 8 = k(2) 2) 3) 3) 4) y = 4x 4) c) When x = 30, y = -6 d) When x = -4, y = 12 1) 1) 2) 2) 3) 3) 4) 4) Page 6 of 12

7 e) When x = 8, y = 80 f) When x = 45, y = -5 The graphs of direct variation equations are always that go through the point (, ). The slope of a direct variation graph is the same value as. Examples: Write an equation for each. y varies directly with x. a) b) Do these graphs show direct variation? If yes, find the slope and write the equation. a) b) c) Page 7 of 12

8 Direct vs Inverse Variation Direct Variation and Inverse Variation both show the relationship between 2 values. Direct Variation Equation y = kx - or - kk = yy xx Inverse Variation k y = x - or - kk = xxxx Example: y = 12 when x = 2 Find k and write the equation for each set of data. 1) y = 3 when x = 15 Direct Variation Inverse Variation 2) when x = -4, y = 20 Page 8 of 12

9 Both forms of variation show a relationship between all input and all output values, therefore the equation represents all solutions for that particular relationship. Direct Variation: yy = kkkk or (kk = yy xx ) Inverse Variation: yy = kk xx or kk = xxxx Create a table of values for each equation. 1) y = 7x 2) 12 y = x 3) y = 1 x 4) 2 24 y = x Determine if each set of data shows direct or indirect variation. Write the equation for each. 5) 6) 7) 8) Page 9 of 12

10 Sketch the graph of each equation. 9) y = 4x 10) 4 y = x Create a table of values for each equation. 11) 16 y = 12) y = 2 x x 3 Determine if each set of data shows direct or indirect variation. Write the equation for each. 13) 14) Page 10 of 12

11 Practice Find the Missing Variable: 1) y varies directly with x. If y = -4 when x = 2, find y when x = -6. 2) y varies inversely with x. If y = 40 when x = 16, find x when y = -5. 3) y varies inversely with x. If y = 7 when x = -4, find y when x = 5. 4) y varies directly with x. If y = 15 when x = -18, find y when x = ) y varies directly with x. If y = 75 when x =25, find x when y = 25. Classify the following as: a) Direct b) Inverse c) Neither 6) m = -5p 9) c = e 4 12) c = 3v 7) r = t 9 10) n = ½ f 13) u = 18 i 8) d = 4t 11) z = What is the constant of variation for the following?.2 t 14) d = 4t 15) z =.2 t 16) n = ½ f 17) r = t 9 Answer the following questions. 18) If x and y vary directly, as x decreases, what happens to the value of y? 19) If x and y vary inversely, as y increases, what happens to the value of x? 20) If x and y vary directly, as y increases, what happens to the value of x? 21) If x and y vary inversely, as x decreases, what happens to the value of y? Page 11 of 12

12 Classify the following graphs as a) Direct b) Inverse c) Neither 22) 23) 24) 25) Answer the following questions: 26) The electric current I, is amperes, in a circuit varies directly as the voltage V. When 12 volts are applied, the current is 4 amperes. What is the current when 18 volts are applied? 27) The volume V of gas varies inversely to the pressure P. The volume of a gas is 200 cm 3 under pressure of 32 kg/cm 2. What will be its volume under pressure of 40 kg/cm 2? 28) The number of kilograms of water in a person s body varies directly as the person s mass. A person with a mass of 90 kg contains 60 kg of water. How many kilograms of water are in a person with a mass of 50 kg? 29) On a map, distance in km and distance in cm varies directly, and 25 km are represented by 2cm. If two cities are 7cm apart on the map, what is the actual distance between them? 30) The time it takes to fly from Los Angeles to New York varies inversely as the speed of the plane. If the trip takes 6 hours at 900 km/h, how long would it take at 800 km/h? Page 12 of 12

6-4 Solving Special Systems

6-4 Solving Special Systems Warm Up Lesson Presentation Lesson Quiz 1 2 pts Bell Quiz 6-4 Solve the equation. 1. 2(x + 1) = 2x + 2 3 pts Solve by using any method. 2. y = 3x + 2 2x + y = 7 5 pts possible