Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
Analyze a transformed reciprocal parent and its asymptotes
Problem
Analyze \(y~=~1/(x~-~2)~+~3\) relative to \(y~=~1/x\). State the parent, transformations, asymptotes, and domain and range exclusions.
Compare the shifted asymptotes and one nearby branch point.Open full size
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What this problem is really about
A reciprocal graph is organized by its two parent asymptotes, so transform those guide lines before thinking about branch points. The denominator’s zero locates the excluded input and vertical boundary, while the outside shift controls the long-run output level. Those same boundaries translate directly into the domain and range exclusions.
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