The graph of an inverse proportion consists of two smooth curves called a hyperbola. Be aware of how it differs from the graph of a direct proportion, which is a straight line. Finally, we will also study application problems involving direct and inverse proportions.
Graphs of Inverse Proportion
The graph of y = a/x is a hyperbola. It does not pass through the origin and does not intersect either the x-axis or the y-axis.
- a ＞ 0: The branches are in the first and third quadrants.
- a ＜ 0: The branches are in the second and fourth quadrants.
- The closer the graph gets to the origin, the closer it gets to the x-axis and y-axis, but it never intersects them (asymptotes).
How to Draw the Graph
① Substitute several values for x, calculate y, and make a table.
② Plot the points (x, y) from the table (as many points as possible).
③ Connect the points with a smooth curve (not a straight line!)
Example: Table for y = 6/x
x：−6, −3, −2, −1, 1, 2, 3, 6
y：−1, −2, −3, −6, 6, 3, 2, 1
An inverse-proportion graph passes through the point (3, −4). Find its equation. Then determine whether this graph passes through the point (−2, 6).
Solution:
Substitute into y = a/x: −4 = a/3 → a = −12. The equation is y = −12/x.
For (−2, 6): xy = (−2)×6 = −12 = a. Since they match, it does pass through the point.
Applications of Direct and Inverse Proportions
Let us organize examples of direct and inverse relationships in everyday life.
・Direct proportion: distance = speed×time (constant speed), cost = unit price×number of items (constant unit price)
・Inverse proportion: time = distance/speed (constant distance), length×width = area of a rectangle (constant area)
Suppose it takes y minutes to walk the 1200 m road from home to the station at a speed of x meters per minute. Express the relationship between x and y as an equation. Then find y when x = 80.
Solution:
Since time = distance / speed, y = 1200/x (inverse proportion).
When x = 80, y = 1200/80 ＝ 15 (minutes).
- The graph of an inverse proportion is a hyperbola (two curves). It does not pass through the origin or intersect the axes.
- a＞0 → first and third quadrants; a＜0 → second and fourth quadrants.
- If speed is constant, the relationship is direct proportion; if distance is constant, the relationship between time and speed is inverse proportion.
- Whether a point (x, y) lies on an inverse-proportion graph can be determined by checking whether it satisfies xy = a.
Practice Problems
- Which quadrants does the graph of the inverse proportion y = −8/x pass through?
- When an inverse-proportion graph passes through (4, 3), find its equation and find y when x = −2.
- Express the relationship between the base area x cm² and height y cm of a rectangular solid with volume 240 cm³. Find y when x = 30.
Answers and Explanations
- Answer: The second and fourth quadrants
Explanation: Since a = −8 ＜ 0, the hyperbola is in the second and fourth quadrants. - Answer: y = 12/x; when x=−2, y = −6
Explanation: a = xy = 4×3 = 12. y = 12/(−2) = −6. - Answer: y = 240/x; when x=30, y = 8
Explanation: Since volume = base area×height, xy = 240 → y = 240/x. 30y = 240 → y = 8.