Problem 17 · 2009 Math Kangaroo
Hard
Algebra & Patterns
transformationsreflection
The diagram illustrates the graphs of the two functions f and g. How can we describe the relationship between f and g?
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Answer: A — \(g(x-2)=-f(x)\)
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Hint 1 of 2
Read off each curve’s vertex: f opens up with its lowest point near x = 1, g opens down.
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Hint 2 of 2
Try flipping f upside-down (that is −f) and sliding it—see which shift lands exactly on g.
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Approach: compare the parabolas as a reflection plus a shift
- f is an upward parabola with vertex near x = 1; g is a downward parabola with vertex near x = −1.
- Reflecting f in the x-axis gives −f, a downward parabola peaking at x = 1; sliding g right by 2 lands its peak at x = 1 too.
- Matching them throughout gives g(x − 2) = −f(x), which is option A.
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