Teach: Direct and Inverse Proportions ② Equation and Properties of Direct Proportion
Direct and Inverse Proportions ② Equation and Properties of Direct Proportion
Two quantities are in direct proportion when one becomes 2 times, 3 times, and so on, and the other also becomes 2 times, 3 times, and so on. Written as an equation, this is y = ax (a is a nonzero constant). It is a basic middle-school function, so begin by understanding direct proportion.
Equation of Direct Proportion
When y is directly proportional to x,
y = ax (a is the constant of proportionality)
- Example: The price y yen of x snacks at 80 yen each → y = 80x. The constant of proportionality is 80.
- Example: When walking at 4 km per hour, the distance y km traveled in x hours → y = 4x. The constant of proportionality is 4.
In a direct proportion, the value of y/x is always constant (= the constant of proportionality a).
Example ①
y is directly proportional to x, and y = 12 when x = 3.
(1) Express y in terms of x.
(2) Find y when x = −5.
Solution:
(1) Substitute x=3 and y=12 into y = ax: 12 = 3a → a = 4. Therefore, y = 4x
(2) When x = −5, y = 4×(−5) = −20
y is directly proportional to x, and y = 12 when x = 3.
(1) Express y in terms of x.
(2) Find y when x = −5.
Solution:
(1) Substitute x=3 and y=12 into y = ax: 12 = 3a → a = 4. Therefore, y = 4x
(2) When x = −5, y = 4×(−5) = −20
Properties of Direct Proportion
- When x is 0, y is also 0 (the graph always passes through the origin).
- When x becomes 2 or 3 times as large, y also becomes 2 or 3 times as large.
- When the sign of x changes, the sign of y also changes (when a＞0).
- y/x ＝ a (a constant)
Example ②
Choose the equations in which y is directly proportional to x, and state the constants of proportionality.
A) y = 3x + 1 B) y = −5x C) y = x² D) y = x/4
Solution:
An equation is a direct proportion if it has the form y = ax.
B) y = −5x (constant of proportionality −5)
D) y = x/4 ＝ (1/4)x (constant of proportionality 1/4)
A) and C) are not direct proportions.
Choose the equations in which y is directly proportional to x, and state the constants of proportionality.
A) y = 3x + 1 B) y = −5x C) y = x² D) y = x/4
Solution:
An equation is a direct proportion if it has the form y = ax.
B) y = −5x (constant of proportionality −5)
D) y = x/4 ＝ (1/4)x (constant of proportionality 1/4)
A) and C) are not direct proportions.
Key Points
- Equation of direct proportion: y = ax (a is the constant of proportionality).
- One pair (x, y) is enough to find a (a = y/x).
- When x = 0, y = 0; in other words, the graph passes through the origin.
- y = ax + b with a constant term is not a direct proportion.
Practice Problems
- y is directly proportional to x, and y = −12 when x = 4. Express y in terms of x.
- For y = 3x, find y when x = −2 and x when y = 9.
- Express the relationship between the walking time x minutes and distance y m for a person walking at 50 m per minute. Also find the distance traveled in 20 minutes.
Answers and Explanations
- Answer: y = −3x
Explanation: Substitute into y = ax: −12 = 4a → a = −3. - Answer: When x = −2, y = −6; when y = 9, x = 3
Explanation: y = 3×(−2) = −6. 9 = 3x → x = 3. - Answer: y = 50x; 1000 m in 20 minutes
Explanation: Distance=speed×time. y=50x. x=20 → y=50×20=1000.
Previous and next lessons in this course
- Previous lesson: Teach: Direct and Inverse Proportions ① Functions and Ranges
- Next lesson: Teach: Direct and Inverse Proportions ③ Graphs of Direct Proportion