Functions — Introduction
GradesAlgebra IFunctions — IntroductionDirect & Inverse Variation

Direct & Inverse Variation

📈 Functions — Introduction
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Direct & Inverse Variation

Identify and write equations for proportional relationships between variables.

Two quantities vary directly if their ratio is constant: as one increases, the other increases proportionally. They vary inversely if their product is constant: as one increases, the other decreases.

Direct Variation

y = kx (k is the constant)y/x = k (constant ratio)Graph: line through originExample: earnings = rate × hours

Inverse Variation

y = k/x (k is the constant)xy = k (constant product)Graph: hyperbolaExample: speed × time = distance
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The constant k is called the constant of variation. Find k by substituting one known (x, y) pair.

✏️Direct Variation: Earnings
Maria earns $15 per hour. When x = hours and y = earnings, y = 15x. Check: y/x = 15 always. If she works 8 hours: y = 15(8) = $120.
✏️Inverse Variation: Speed & Time
A 120-mile trip: y = 120/x where x = speed and y = time. At 40 mph, t = 3 hours. At 60 mph, t = 2 hours. Product xy = 120 stays constant.

Writing a Variation Equation

1Identify whether the relationship is direct (y = kx) or inverse (y = k/x).
2Substitute the given point (x, y) to find k.
3Write the complete equation with the value of k.
4Use the equation to find missing values.
✏️ Try It!

If y varies directly with x, and y = 24 when x = 6, what is y when x = 10?

Practice with CalcVerse
⚖️Unit Rate & Ratio Solver
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