Which of the following could represent the graph of $y = -x^2 + bx + 1$, where $b > 0$?
A - HSC - SSCE Mathematics Advanced - Question 7 - 2020 - Paper 1
Question 7
Which of the following could represent the graph of $y = -x^2 + bx + 1$, where $b > 0$?
A.
B.
C.
D.
Worked Solution & Example Answer:Which of the following could represent the graph of $y = -x^2 + bx + 1$, where $b > 0$?
A - HSC - SSCE Mathematics Advanced - Question 7 - 2020 - Paper 1
Step 1
Identify the equation
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Answer
We are given the equation of a parabola in the form y=−x2+bx+1. The negative coefficient of x2 indicates that the parabola opens downwards.
Step 2
Determine the vertex
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Answer
To find the vertex of the parabola, we can use the formula for the x-coordinate of the vertex, given by x=−2ab. Here, a=−1 and b is positive, hence the parabola will peak at the vertex. Since b>0, the vertex will be to the right of the origin.
Step 3
Analyze the options
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Answer
The graphs need to be examined for the following features:
The graph should open downwards (which eliminates any graphs that open upwards).
The vertex will be above the x-axis since the constant term is +1.
The parabola should intersect the y-axis at 1.
Given these characteristics, option A matches these criteria.
