Cogito
Algebra 2 · Chapter 4 · Lesson 3
Direct, Inverse, and Joint Variation
Two quantities, one constant holding them together.
12 problems · about 21 minutes · A-CED.A.2, A-CED.A.4
What this lesson teaches
The student writes and applies direct, inverse and joint variation models.
- Direct variation is y = kx; inverse variation is y = k ÷ x.
- One matching pair of values is enough to find k.
- Joint variation depends on two quantities at once, as in y = kxz.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy varies inversely with x, and y = 10 when x = 3. What is k?
Answer 30
Why 30.
Which graph belongs to direct variation?
Answer A line through the origin.
Why A line through the origin.
y varies directly with x, and y = 20 when x = 4. What is k?
Answer 5
Why 20 ÷ 4.
Same relationship. What is y when x = 7?
Answer 35
Why 5 × 7.
y varies inversely with x, and y = 6 when x = 4. What is k?
Answer 24
Why 6 × 4.
Build It Up
The same ideas with more to keep track of.
3 problemsSame relationship. What is y when x = 8?
Answer 3
Why 24 ÷ 8.
y varies jointly with x and z, with k = 2. What is y when x = 3 and z = 5?
Answer 30
Why 2 × 3 × 5.
Pressure varies inversely with volume, and k = 100. What is the pressure when the volume is 25?
Answer 4
Why 100 ÷ 25.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each pairing by the kind of variation it shows.
Answer Direct: x = 2, y = 6 and x = 4, y = 12, Hours worked and total pay · Inverse: x = 2, y = 12 and x = 4, y = 6, Speed and travel time over a fixed distance
Why Direct rises together; inverse trades one for the other.
y varies directly with x. When x = 9, y = 27. What is x when y = 12?
Answer 4
Why k = 3, so x = 12 ÷ 3.
The Wages: Pay varies directly with hours. 6 hours pays $90. What does 10 hours pay, in dollars?
Answer 150 dollars
Why $150.
The Workers: Time varies inversely with the number of workers. 4 workers take 12 days. How many days do 6 workers take?
Answer 8 days
Why 8 days.