Direct, Inverse, and Joint Variation
Identify what stays constant before substituting numbers or comparing changes.
Direct, Inverse, and Joint Variation
Not yet verified on this browser.
Variation describes how quantities change together
Direct variation keeps a ratio constant; inverse variation keeps a product constant; joint variation relates one quantity to the product of two or more others.
direct: y = kxinverse: y = kxjoint: z = kxyCombined variation may place some quantities in the numerator and others in the denominator.
Find k once
Use one complete set of known values to determine the constant of variation, then reuse the model.
Compare factors when possible
If only relative change is asked, multiply the change factors instead of solving for every original value.
Use ratio for direct, product for inverse
Check a table quickly: constant yx suggests direct variation; constant xy suggests inverse variation. For combined variation, write the numerator quantities first and place inverse quantities in the denominator.
Why it works
The constant k captures everything in the relationship that does not change. Once k—or the relative scale factor—is known, new values follow predictably.
Five forms you should recognize
Problem: y varies directly as x; y = 18 when x = 12. Find y when x = 30.
k = 1812 = 1.5; y = 1.5(30) = 45Why: Direct variation keeps yx constant.
Problem: y varies inversely as x; y = 24 when x = 5. Find y when x = 8.
xy = 120; y = 1208 = 15Why: Increasing x requires y to decrease so the product remains constant.
Problem: z varies jointly as x and y; z = 60 when x = 5,y = 4.
60 = k(5)(4) → k = 3Why: Joint variation multiplies the contributing quantities.
Problem: y varies directly as x and inversely as z².
y = kxz²Why: Direct quantities belong in the numerator; inverse quantities belong in the denominator.
Problem: V varies as r²h. Double r and halve h.
neworiginal = 2²(12) = 2Why: Apply each variable’s change factor with its stated exponent.
Check before you commit
- Calling any increasing relationship a direct variation
- Ignoring a fixed starting value that prevents direct proportionality
- Using a constant ratio for an inverse relationship
- Forgetting an exponent in a combined-variation model
- Saying “decreases by one-third” when the new value is one-third of the original
- Solving for k when a quick change-factor comparison is enough
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Direct, Inverse, and Joint Variation FAQ
How can I recognize direct variation from a graph?
Its graph is a straight line through the origin.
How can I recognize inverse variation from a table?
The product of corresponding x and y values remains constant.
Do I always need to calculate k?
No. For relative-change questions, comparing scale factors is often faster.
Continue your mathematics review.
Your progress stays on this browser.
Mastery results save to your Teacher Abi study profile.
Return to Student Hub View UPCAT Coverage
No comments:
Post a Comment