Question
The diagram shows part of the graph of \(y=a+b\sin x\). State the values of the constants a and b.
Answer/Explanation
a=1,b=2
Question
(a)The diagram shows part of the graph of \(y=a+b\sin x\).Find the values of constant a and b.
(b)(i)Show that the equation \(\left ( \sin \Theta +2\cos \Theta \right )\left ( 1+\sin \Theta -\cos \Theta \right )=\sin \Theta \left ( 1+\cos \Theta \right )\)
may be expressed as \(3\cos ^{2}\Theta -2\cos \Theta -1=0\).
(iii)Hence solve the equation \(\left ( \sin \Theta +2\cos \Theta \right )\left ( 1+\sin \Theta-\cos \Theta \right )=\sin \Theta \left ( 1+\cos \Theta \right )\) for \(-180^{\circ}\leq \Theta \leq 180^{\circ}\)
Answer/Explanation
(i)a=-2,b=3
(b)(ii)\(s+s^{2}-sc+2c+2sc=s+sc\rightarrow s^{2}-2c^{2}+2c=0\)
\(1-\cos ^{2}\Theta -2\cos ^{2}\Theta +2\cos \Theta =0\)
\(3\cos ^{2}\Theta -2\cos \Theta -1=0\)
(ii)\(\cos \Theta = 1\)or \( -\frac{1}{3}\)
\(\Theta =0^{\circ}\) or \(109.5^{\circ}\) or \(-109.5^{\circ}\)
Question.
The function f is defined by \(f: x\rightarrow 4 sin x – 1\) for \(\frac{-\Pi}{2}\leq x\leq\frac{\Pi}{2}\).
(i) State the range of f.
(ii) Find the coordinates of the points at which the curve y = f(x) intersects the coordinate axes.
(iii) Sketch the graph of y = f(x).
(iv) Obtain an expression for \(f^{-1}(x)\) , stating both the domain and range of \(f^{-1}\).
Answer/Explanation
(i) \(f:x\rightarrow 4\sin x-1\) for \(-\frac{\pi }{2}\leq x\leq \frac{\pi }{2}\)
Range \(-5\leq f(x)\leq 3\)
(ii) 4s-1=0→\(s=\frac{1}{4}\rightarrow x=0.253\)
\(x=0\rightarrow y=-1\)
(iv)range-\(\frac{1}{2}\pi \leq f^{-1}\left ( x \right )\leq\frac{1}{2}\pi \)
domain \(-5\leq x\leq 3\)
Inverse \(f^{-1}(x)=\sin ^{-1}\left ( \frac{x+1}{4} \right )\)
Question
(a) Solve the equation \(3sin^{2}2\Theta+8cos2\Theta =0\) for 0Å ≤ 1 ≤ 180Å.
(b)
The diagram shows part of the graph of y = a + tan bx, where x is measured in radians and a and
b are constants. The curve intersects the x-axis at \( (-\frac{\Pi }{ 6},0) \)and the y-axis at \((0,\sqrt{3}) \) Find the
values of a and b.
Answer/Explanation
(a)\(3(1-cos^{2}2\Theta)+8cos2\Theta =0\rightarrow 3cos^{2}2\Theta -8cos2\Theta -3(=0)\)
cos2θ\(=-\frac{1}{3}\)
2θ \(= 109.(47)o or 250.(53)o\)
θ = 54.7o or 125.3o
(b)
√3 tan0 3 = + a , a →=√3
0 tan( −bπ/ 6) +√ 3 taken as far as \(tan^{-1}\), angle units consistent
b=2
Question
In the diagram, the lower curve has equation y = cos θ. The upper curve shows the result of applying
a combination of transformations to y = cos θ.
Find, in terms of a cosine function, the equation of the upper curve. [3]
Answer/Explanation
Ans
4 \((y=)[3]+[2]\left [ \cos \frac{1}{2}\theta \right ]\)
Question
In the diagram, the graph of y = f(x) is shown with solid lines. The graph shown with broken lines is a transformation of y = f(x).
(a) Describe fully the two single transformations of y = f(x) that have been combined to give the resulting transformation.
(b) State in terms of y, f and x, the equation of the graph shown with broken lines.
Answer/Explanation
Ans:
a)(Stretch)(factor 3 in y direction or parallel to the y-axis)
(Translation)\(\binom{4}{0}\)
b) [y=]3f(x-4)
Question.
The diagram shows part of the graph of y = a cos (bx) + c.
(a) Find the values of the positive integers a, b and c.
(b) For these values of a, b and c, use the given diagram to determine the number of solutions in the interval 0 ≤ x ≤ 2π for each of the following equations.
(i) \( a cos (bx) + c =\frac{6}{\pi }x\)
(ii) \(a cos (bx) + c = 6- \frac{6}{\pi }x\)
Answer/Explanation
(a) \(a = 5, b = 2 , c = 3\)
b.(i) 3
b.(i) 2
Question
The diagram shows part of the graph of y = a tan (x − b) + c.
Given that 0 < b < π, state the values of the constants a, b and c
Answer/Explanation
Ans
\(a = 2\)
\(b=\frac{\pi }{4}\)
\( c = 1\)
Question
(a)Express 2x2 − 8x + 14 in the form 2[(x – a)2 + b]
The functions f and g are defined by
f(x) = x2 for x ∈ R >,
g(x) = 2x2 − 8x + 14 for x ∈ R>.
(b) Describe fully a sequence of transformations that maps the graph of y = f(x) onto the graph of y= g(x), making clear the order in which the transformations are applied.
Answer/Explanation
(a) \(2\left [ \left \{ \left ( x-2 \right )^{2} \right \}\left \{ +3 \right \} \right ]\)
(b){Translation} \( \binom{\left \{ 2 \right \}}{\left \{ 3 \right \}}\) OR {Stretch} {y direction} {factor 2}
{Stretch} {y direction} {factor 2} OR {Translation} \( \binom{\left \{ 2 \right \}}{\left \{ 6 \right \}}\)