Direct proportion is the relationship where, when one quantity doubles or triples, the other doubles or triples too; it is written y = ax.
Walking at a steady speed is a good example: walk for twice as long and you cover twice the distance.
Inverse proportion has the form y = a/x: over a fixed distance, doubling your speed halves the time, so as one quantity grows the other shrinks and their product xy stays the same.
On this page you change the constant a to see a straight line through the origin and a hyperbola, and check in the pictures whether the quotient y ÷ x or the product x × y stays fixed.
Direct proportion, y = ax
seegongsik.com
Direct proportion: a line through the origin, with the points at x = 1, 2, 3
2
👀 Direct proportion
①If y doubles, triples, … whenever x doubles, triples, …, then y is directly proportional to x
②The equation has the form y = ax (a ≠ 0), and for any non-zero x, y ÷ x is always a
③The graph is a straight line through the origin. The three points in the picture all lie on one line
④If a > 0 the line rises to the right; if a < 0 it falls to the right
Inverse proportion, y = a/x
seegongsik.com
Inverse proportion: a hyperbola, with rectangles that all have area a
6
🔍 Inverse proportion
①If y becomes ½, ⅓, … whenever x doubles, triples, …, then y is inversely proportional to x
②The equation has the form y = a/x (a ≠ 0), and x × y is always a. That is why the rectangles in the picture all have the same area
③The graph is a pair of curves that get ever closer to the axes without touching them: a hyperbola
④If a > 0 it lies in quadrants I and III; if a < 0, in quadrants II and IV
What the constant means
Constant of direct proportion
a = yx
For x ≠ 0, y ÷ x is always a
Constant of inverse proportion
a = xy
x × y is always a
💡 Finding the constant
①For direct proportion, one point (x, y) on the graph is enough: a = y ÷ x
②For inverse proportion, one point (x, y) is enough too: a = x × y
③The sign of a decides which quadrants the graph passes through
④In direct proportion, the larger |a| is, the closer the line is to the y-axis and the steeper it gets
Check with examples
Example 1
y is directly proportional to x, and y = 6 when x = 2. Write y in terms of x.
1
Direct proportion has the form y = ax. Put in x = 2 and y = 6 to find a.
6 = a × 2 ⇒ a = 3
2
Put a = 3 into the equation.
y = 3x
▸ y = 3x
Set a to 3 in the picture and the points for x = 1, 2, 3 fall at (1, 3), (2, 6), (3, 9); 9 is off the grid, so only the first two show.
Example 2
y is inversely proportional to x, and y = 4 when x = 3. Find y when x = 6.
1
Inverse proportion has the form y = a/x, and a = x × y = 3 × 4 = 12.
y = 12/x
2
Put in x = 6.
y = 12/6 = 2
▸ y = 2
When x doubled from 3 to 6, y halved from 4 to 2. The product stays the same: 3 × 4 = 6 × 2 = 12.
Exam checklist
Direct proportion
y = ax → a straight line through the origin
The larger |a|, the steeper the line and the closer it is to the y-axis
Inverse proportion
y = ax
a > 0: quadrants I and III; a < 0: quadrants II and IV
Grade 7 school test style
The graph of the inverse proportion y = 12/x passes through the point (a, 3). What is a?
①2
②3
③4
④6
⑤12
▸ ③ 4
1
Put the point (a, 3) into the equation.
3 = 12/a
2
Solve for a.
a = 12 ÷ 3 = 4
🎯 Exam checklist
①Tell direct and inverse proportion apart by whether y ÷ x or x × y stays fixed
②Judge the sign of the constant from the quadrants the graph passes through
③In direct proportion, a larger absolute value of a means a steeper line
④The graph of inverse proportion gets ever closer to the axes but never meets them
⑤At a fixed speed, distance and time are directly proportional; over a fixed distance, speed and time are inversely proportional