Question

The diagram shows part of the graph of $$y=a+b\sin x$$. State the values of the constants a and b.

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 $$-180^{\circ}\leq \Theta \leq 180^{\circ}$$

(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 $$f^{-1}$$.

(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.

(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]

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.

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)   $$a cos (bx) + c = 6- \frac{6}{\pi }x$$

(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

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.

(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}  $$\binom{\left \{ 2 \right \}}{\left \{ 6 \right \}}$$