Problem 17 · 2011 Math Kangaroo
Hard
Algebra & Patterns
evaluate-formulasubstitution
In the (x, y)-plane the coordinate axes are drawn as usual. The point A(1, −10), which lies on the parabola \(y=ax^{2}+bx+c\), was marked. Then the coordinate axes and most of the parabola were erased, leaving the sketch on the right. Which of the following statements could be false?
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Answer: E — \(c<0\)
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Hint 1 of 2
Use the point A(1, −10): substituting x = 1 gives a + b + c = −10.
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Hint 2 of 2
That fixes some statements as always true; look for the one that the picture does not force.
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Approach: substitute the known point and test each statement
- A(1,−10) gives a+b+c = −10 < 0, so that statement is always true; the upward shape forces a > 0.
- But c is the value at x = 0, which the trimmed picture does not pin down — it may be positive.
- Hence c < 0 is the statement that could be false.
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