The diagram below shows the graph with equation $y = f(x)$, where
$f(x) = k(x-a)(x-b)$ - Scottish Highers Maths - Question 15 - 2016
Question 15
The diagram below shows the graph with equation $y = f(x)$, where
$f(x) = k(x-a)(x-b)$.
(a) Find the values of $a$, $b$ and $k$.
(b) For the function $g(y) = f(x)... show full transcript
Worked Solution & Example Answer:The diagram below shows the graph with equation $y = f(x)$, where
$f(x) = k(x-a)(x-b)$ - Scottish Highers Maths - Question 15 - 2016
Step 1
Find the values of $a$
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Answer
From the graph, the roots are located at x=−5 and x=4. Based on the positive root structure of the equation, we assign:
a=−5
Step 2
Find the values of $b$
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Answer
Thus, we also identify:
b=4
Step 3
calculate $k$
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Answer
To find the value of k, we use the known point (1,9).
Substituting into the equation:
f(1)=k(1−a)(1−b)
We calculate:
f(1)=k(1−(−5))(1−4)9=k(6)(−3)9=−18k
Solving for k yields:
k = - rac{1}{2}
Step 4
For the function $g(y) = f(x) - d$, determine the range of values of $d$
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Answer
For g(y) to have exactly one real root, d must be equal to the maximum value of f(x). From the graph, the maximum value of f(x) occurs at the vertex, which is at (1,9):
Thus, the only value for d that allows g(y) to have exactly one real root is:
d<9
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