Consider the polynomial $p(x) = ax^3 + bx^2 + cx - 6$ with $a$ and $b$ positive - HSC - SSCE Mathematics Extension 1 - Question 10 - 2016 - Paper 1
Question 10
Consider the polynomial $p(x) = ax^3 + bx^2 + cx - 6$ with $a$ and $b$ positive.
Which graph could represent $p(x)$?
(A)
(B)
(C)
(D)
Worked Solution & Example Answer:Consider the polynomial $p(x) = ax^3 + bx^2 + cx - 6$ with $a$ and $b$ positive - HSC - SSCE Mathematics Extension 1 - Question 10 - 2016 - Paper 1
Step 1
Determine the Behavior of the Polynomial
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Answer
Since a>0, the leading coefficient of p(x) is positive. This means that as x→+∞, p(x)→+∞ and as x→−∞, p(x)→−∞. Therefore, we expect the polynomial to start low and rise as x increases.
Step 2
Identify the Number of Turning Points
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Answer
A cubic polynomial can have up to 2 turning points. The graph should exhibit one or two local extrema, which means we expect it to change direction at least once.
Step 3
Analyze the Given Options
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Answer
Next, we evaluate the graphs:
Graph A: Starts low, rises toward the right, and has one turning point, fitting the expected behavior.
Graph B: Appears bounded on both ends and does not match the polynomial’s behavior.
Graph C: Starts low with a single turning point and ends at the same height, which is inconsistent for a cubic.
Graph D: Has a minimum and a maximum, but does not start low and rise high as it moves to the right.
Thus, only Graph A matches our analysis.
Step 4
Conclusion
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Answer
Based on the analysis of end behavior and turning points, the suitable graph to represent the polynomial p(x) is Graph A.
