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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
Identify the Behavior of the Polynomial
Answer
Given that and are positive, the leading term suggests that as approaches positive infinity, will also approach positive infinity, and as approaches negative infinity, will approach negative infinity. This implies that the graph of will start low (negative) on the left side and rise to the right.
Step 2
Examine the Graphs
Answer
Now, we must analyze each graph option:
- Graph A: Starts low and rises, which matches our derived behavior.
- Graph B: Appears flat or has asymptotic behavior, not matching.
- Graph C: Starts high (positive) on the left, which contradicts our findings.
- Graph D: Displays a similar shape to Graph A but may not fit based on further inspection of turning points.
Only Graph A is consistent with the behavior of as derived.