In the diagram below, the graph of $f(x) = an(x - 45^ ext{o})$ is drawn for the interval $x ext{ } ext{ } ext{for } [ -90^ ext{o}, 180^ ext{o}].$
6.1 Write down the period of $f$ - NSC Mathematics - Question 6 - 2023 - Paper 2
Question 6
In the diagram below, the graph of $f(x) = an(x - 45^ ext{o})$ is drawn for the interval $x ext{ } ext{ } ext{for } [ -90^ ext{o}, 180^ ext{o}].$
6.1 Write down... show full transcript
Worked Solution & Example Answer:In the diagram below, the graph of $f(x) = an(x - 45^ ext{o})$ is drawn for the interval $x ext{ } ext{ } ext{for } [ -90^ ext{o}, 180^ ext{o}].$
6.1 Write down the period of $f$ - NSC Mathematics - Question 6 - 2023 - Paper 2
Step 1
Write down the period of $f$
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Answer
The period of the function f(x)=an(x−45exto) is 180exto.
Step 2
Draw the graph of $g(x) = - ext{cos}(2x)$ for the interval $x ext{ for } [-90^ ext{o}, 180^ ext{o}]$
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Answer
The graph of g(x) is a cosine wave reflected over the x-axis, with its amplitude equal to 1 and a period of 90exto. It will intercept the x-axis at multiples of 45exto and reach its maximum at x=−90exto and minimum at x=0, shown clearly across the specified range.
Step 3
Write down the range of $g$
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Answer
The range of g(x)=−extcos(2x) is between [−1,1], inclusive.
Step 4
Determine the equation of $h$ in its simplest form
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Answer
If g(x) is shifted 45exto to the left, the equation becomes h(x)=−extcos(2(x+45exto))=−extcos(2x+90exto). This simplifies to h(x)=extsin(2x).
Step 5
Use the graph(s) to determine the values of $x$ in the interval $[-90^ ext{o}, 90^ ext{o}]$ for which $f(x) > 1$
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Answer
The values of x for which f(x)>1 can be found in the interval xextfor[−90exto,−45exto)extand(45exto,90exto].
Step 6
Use the graph(s) to determine the values of $x$ in the interval $[-90^ ext{o}, 90^ ext{o}]$ for which $2 ext{cos}(2x) - 1 > 0$
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Answer
Solving 2extcos(2x)−1>0 gives ext{cos}(2x) > rac{1}{2}, which corresponds to 2xextin[−60exto,60exto]. Therefore, the values of x in the specified interval are in [−30exto,30exto].
![In-the-diagram-below,-the-graph-of-$f(x)-=--an(x---45^-ext{o})$-is-drawn-for-the-interval-$x--ext{-}--ext{-}--ext{for-}-[--90^-ext{o},-180^-ext{o}].$--6.1-Write-down-the-period-of-$f$-NSC Mathematics-Question 6-2023-Paper 2.png](https://cdn.simplestudy.cloud/assets/backend/uploads/question/subject_692_paper_20454_q6_679949c8.jpg)
