Reciprocal graphs
A nonzero fixed product gives two branches when all allowed real inputs are included. Physical models may use only the positive branch
- Fixed product Use xy=a to connect a table or marked point with the rule. Nonzero inputs of either sign are allowed
- Excluded values The unshifted curve excludes x=0 and never reaches y=0. Its asymptotes are the coordinate axes
- Signs and scale Positive a gives branches in quadrants I and III; negative a gives branches in II and IV. Doubling a doubles every output at a fixed input
- Translations For a translated reciprocal rule, the vertical asymptote has input h and the horizontal asymptote has output k. The intersection of these lines is not on the curve
- Equations and intersections Check excluded inputs and both branches. A target equal to the horizontal asymptote is never reached. A line may meet neither, one or both branches, depending on its rule
- Physical domain A rectangle requires positive dimensions. State the fixed-area assumption and check measurements and allowances
Worked example
Solve the shifted reciprocal equation and verify the input is allowed
- \(\displaystyle \frac{-6}{x+1}=-3\)Subtract the outside translation 2.
- \(\displaystyle x+1=2\)The denominator must be nonzero. Divide −6 by −3.
- \(\displaystyle x=1\)The denominator is 2, and the original output is −3+2=−1.
Common mistakes
Difficulty levels
Questions about this topic
Why are there two branches?+
Inputs on either side of the excluded value give points, but the undefined input separates them. Never connect across the gap
Can a reciprocal curve cross an axis?+
An unshifted y=a/x curve cannot. A shifted curve may cross a coordinate axis, but still cannot cross its own vertical or horizontal asymptote
Is it decreasing everywhere?+
For positive a, it decreases on each side of the vertical asymptote separately. Across the excluded input, a global decreasing claim is false
Is every decreasing curve inverse proportion?+
No. For inverse proportion, check that xy stays constant. A shifted reciprocal need not have a constant product xy
Why can a graph intersection be missed?+
A narrow viewing window may show only one branch or crossing. Use the rule to find all candidates, check excluded values and adjust the window
Are negative widths meaningful?+
No for a physical rectangle. Distinguish the unrestricted mathematical curve from the positive domain of that model
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs