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The diagram shows the graph of $y = a(x + b)(x + c)(x + d)^2$ - HSC - SSCE Mathematics Extension 1 - Question 6 - 2018 - Paper 1
Question 6
The diagram shows the graph of $y = a(x + b)(x + c)(x + d)^2$. What are possible values of a, b, c and d? A. a = -6, b = -2, c = -1, d = 1 B. a = -6, b = 2,... show full transcript
Worked Solution & Example Answer:The diagram shows the graph of $y = a(x + b)(x + c)(x + d)^2$ - HSC - SSCE Mathematics Extension 1 - Question 6 - 2018 - Paper 1
Step 1
What are possible values of a, b, c and d?
Answer
To determine the possible values of a, b, c, and d, we need to analyze the given polynomial equation and the characteristics of the graph provided.
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Identifying Roots:
The graph crosses the x-axis at three points, indicating that there are roots at these coordinates. Since one of the roots has a multiplicity of 2, we can infer that at least one of the factors in the expression corresponds to this multiplicity. -
Analyzing the Graph:
To match the graph's shape and its turning points, we deduce the appropriate candidates for b, c, and d accordingly. -
Evaluating Options:
- For option A: The coefficients suggest all roots are negative and could work.
- For option B: This would not create the appropriate graph due to the positive nature of b.
- For option C: This could match the roots and behavior of the graph.
- For option D: This fails to satisfy the conditions needed for the roots of the polynomial.
Based on this analysis, option C is the only valid pair that aligns with the graph's characteristics.