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Direct variation y=kx has constant ratio xy; inverse variation y=xk has constant product xy.
Worked example 1
01 / Classify the relationships represented by A=7t, B=t84, and C=7t+4. State an invariant for each variation model and its input restriction.Direct line A = 7t passes through the origin; affine line C = 7t + 4 does not. Separately classify D=t+184 as a relation to t, state its domain, and justify why a shifted reciprocal is not ordinary inverse variation.
02 / The direct model is a constant times the input, so its graph passes through the origin.
03 / The inverse model has constant input-output product and is undefined at zero.
04 / The nonzero intercept in C prevents a constant ratio, and multiplying C by t does not give a constant product.
05 / The shifted reciprocal has domain t=−1. At inputs one and two its outputs are forty-two and twenty-eight. The ratios forty-two and fourteen differ, and the products forty-two and fifty-six differ, so it is neither direct nor inverse variation in the stated input.
Worked example 2 · Recognize direct and inverse variation
Data pairs are (2,18), (3,12), and (6,6). Decide whether direct or inverse variation fits all three and explain what the finite data do not establish.
The inverse curve y = 36/x passes through all three supplied data points.
The products are 2⋅18=3⋅12=6⋅6=36. The ratios are 9, 4, and 1, so direct variation does not fit. The inverse model passes through every supplied point, but unsampled behavior needs a modeling assumption or further evidence.
Read the graphs
Recognize direct and inverse variation
Classify the relationships represented by A=7t, B=t84, and C=7t+4. State an invariant for each variation model and its input restriction.Direct line A = 7t passes through the origin; affine line C = 7t + 4 does not. Separately classify D=t+184 as a relation to t, state its domain, and justify why a shifted reciprocal is not ordinary inverse variation.
The direct model is a constant times the input, so its graph passes through the origin. The inverse model has constant input-output product and is undefined at zero. The nonzero intercept in C prevents a constant ratio, and multiplying C by t does not give a constant product. The shifted reciprocal has domain t=−1. At inputs one and two its outputs are forty-two and twenty-eight. The ratios forty-two and fourteen differ, and the products forty-two and fifty-six differ, so it is neither direct nor inverse variation in the stated input.
Answer: A is direct, defined for all real t, with ratio tA=7 when t=0; B is inverse with Bt=84 and t=0; C is neither. The additional model D is neither, with t=−1.
Determine and interpret the variation constant
The mass of uniform wire varies directly with length. A7.5m piece has mass 3 kg. Find and interpret the constant, then write the model.Mass of uniform wire versus its nonnegative length. In your written work, sketch the mass-versus-length graph on its physical domain and interpret its slope.
Write M=kL before substituting data. Use 3=7.5k, giving k=7.53=0.4. The constant is mass per unit length, so its units are kilograms per meter. The sketch is a ray through the origin with horizontal length in meters and vertical mass in kilograms, for nonnegative length. Its slope 0.4 kilogram per meter is mass per unit length.
Answer: k=0.4 kg per m; M=0.4L.
Model joint and inverse-square relationships
Light from an ideal point source spreads uniformly over spheres. Explain the inverse-square distance model and predict the intensity at 6m if it is 90 units at 2m, with power fixed. Sketch intensity against positive distance, label axes, and mark the two stated distances and their intensities.Use the labeled scale, curve styles, and domain information.
A sphere of radius r has area 4πr2. For fixed emitted power, intensity is power divided by area, so it varies inversely as r2. Tripling radius multiplies area by nine and divides intensity by nine: 990=10. The graph is I=r2360 on r>0. It passes through (2,90) and (6,10), decreases on its positive-distance domain, and never reaches zero at a finite positive distance.
Answer: 10 units at 6m; sphere area scales as r2.
Choose meaningful variables and test modeling assumptions
A spring has natural length 6 cm. A force of 45N maintains a total length of 9 cm. Under the direct-extension model, find the spring constant and the force for total length 11 cm. Sketch force against extension and force against total length, label axes and intercepts, and explain which is direct variation.Use the labeled scale, curve styles, and domain information.Use the labeled scale, curve styles, and domain information.
Extension is the amount beyond natural length: x=L−6. The reference extension is 9−6=3 cm, so 45=3k gives k=15N per cm. At total length eleven, extension is five, so F=15(11−6)=75N. The extension graph is the ray F=15x, x≥0, through the origin. The total-length graph is F=15(L−6), L≥6, beginning at (6,0). Label both physical axes and points. An offset in total length prevents direct variation in that variable.
Answer: k=15N per cm; force 75N. Force varies directly with extension, not total length.
15 questions covering every idea, with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.