Home / IBDP Maths analysis and approaches Topic: SL 3.7 :Composite functions of the form HL Paper 1

IBDP Maths analysis and approaches Topic: SL 3.7 :Composite functions of the form HL Paper 1

Question

The diagram below shows a curve with equation \(y = 1 + k\sin x\) , defined for \(0 \leqslant x \leqslant 3\pi \) .

 

 

The point \({\text{A}}\left( {\frac{\pi }{6}, – 2} \right)\) lies on the curve and \({\text{B}}(a,{\text{ }}b)\) is the maximum point.

(a)     Show that k = – 6 .

(b)     Hence, find the values of a and b .

▶️Answer/Explanation

Markscheme

(a)     \( – 2 = 1 + k\sin \left( {\frac{\pi }{6}} \right)\)     M1

\( – 3 = \frac{1}{2}k\)     A1

\(k = – 6\)     AG     N0

(b)     METHOD 1

maximum \( \Rightarrow \sin x = – 1\)     M1

\(a = \frac{{3\pi }}{2}\)     A1

\(b = 1 – 6( – 1)\)

\( = 7\)     A1     N2

METHOD 2

\(y’ = 0\)     M1

\(k\cos x = 0 \Rightarrow x = \frac{\pi }{2},{\text{ }}\frac{{3\pi }}{2},{\text{ }} \ldots \)

\(a = \frac{{3\pi }}{2}\)     A1

\(b = 1 – 6( – 1)\)

\( = 7\)     A1     N2

Note: Award A1A1 for \(\left( {\frac{{3\pi }}{2},{\text{ }}7} \right)\) .

[5 marks]

Question

The following diagram shows the curve \(y = a\sin \left( {b(x + c)} \right) + d\), where \(a\), \(b\), \(c\) and \(d\) are all positive constants. The curve has a maximum point at \((1,{\text{ }}3.5)\) and a minimum point at \((2,{\text{ }}0.5)\).

a. Write down the value of \(a\) and the value of \(d\).[2]

b.Find the value of \(b\).[2]

c.Find the smallest possible value of \(c\), given \(c > 0\). [2]

▶️Answer/Explanation

Markscheme

a. \(a = 1.5\,\,\,d = 2\)    A1A1

[2 marks]

b.

\(b = \frac{{2\pi }}{2} = \pi \)    (M1)A1

[2 marks]

c.

attempt to solve an appropriate equation or apply a horizontal translation     (M1)

\(c = 1.5\)    A1

Note:     Do not award a follow through mark for the final A1.

Award (M1)A0 for \(c =  – 0.5\).

[2 marks]

Question

Sketch the graphs of the functions
(a) f(x)=|sin 4x|,  0≤x≤\(\pi\)
(b) f(x)=|sin(\(\pi x\))|, 0≤x≤4
(c) f(x)=sin|4x|, -\(\pi\)≤x≤\(\pi\)

▶️Answer/Explanation

Ans
(a) f(x) = |sin 4x|, 0≤x≤\(\pi\)

(b) f(x) =|sin(\(\pix\))|, 0≤x≤4

(c) f(x)=sin|4x|,   -\(\pi\)≤x≤\(\pi\)

Question

Consider the function f(x) = e2x and g(x) = sin\(\frac{\pi x}{2}\).
(a) Find the period of the function fºg
(b) Find the intervals for which (fºg(x))>4

▶️Answer/Explanation

Ans
(a) fºg(x)=\(e^{2sin(\frac{\pi x}{2})}\)
Period = period of sin (\(\frac{\pi x}{2}\))
= 4   (Accept \(\frac{720o}{\pi}=229o\))
(b)
\(x\epsilon (0.488+4k,1.51+4k)(4\epsilon \mathbb{Z})(Accept x\epsilon (28.0^o+\frac{720^o}{\pi}k,86.5^o+\frac{720^o}{\pi}k))\)

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