Let $f(x)=x^{3}$ and $g(x)=5(x-2)^{3}$ , for $x \in \mathbb{R}$ .
The graph of g can be obtained from the graph of f using two transformations.
1. Give a full geometric description of each of the two transformations.

The graph of g is translated by the vector $\binom{4}{-1}$ to give the graph of a function h . The point $\mathrm{P}(1,1)$ on the graph of f is translated to point $\mathrm{Q} $ on the graph of h .
2. Find the coordinates of $\mathrm{Q}$ .

Let f and g be functions such tha+ww rji1jn7o 1t $g(x)=3 f(x-2)+1$ , for $x \in \mathbb{R}$ .
The graph of g is obtained from the graph of f after the following transformations:
a vertical stretch by a factor of k , followed by
a translation by the vector $\binom{a}{b}$ .
1. Write down the value of:
1. k ;
2. a ;
3. b .

Let h(x)=-2 g(x) , for $x \in \mathbb{R}$ .
The point $\mathrm{P}(3,4)$ on the graph of g is mapped to point $\mathrm{Q}$ on the graph of h .
2. Find the coordinates of Q .

Let $f(x)=0.2 e^{x+2}-4$ , for $-3 \leq x \leq 2 $.
1. On the following grid, sketch the graph of y=f(x) .

2. Find the coordinates of:
1. the x -intercept;
2. the y -intercept.

The graph of f is reflected in the x -axis and then translated by the vector $\binom{1}{2}$ to obtain the graph of a function g .
3. Find g(x) .

$\text { The following diagram shows the graph of } y=f(x) \text {, for }-1 \leq x \leq 2 \text {. }$



1. Write down the value of:
1. f(1) ;
2. $ f^{-1}(-2)$ .
2. Find $(f \circ f)(1) $.
3. Sketch the graph of y=f(-x) on the same grid above.

  Let $f(t)=3 t^{2}+27$ , where t>0 .
The graph of a function g is obtained when the graph of f is transformed by
a stretch by a scale of $\frac{1}{9}$ parallel to the y -axis, followed by a translation by the vector $\binom{4}{-5}$ .
1. Find g(t) , giving your answer in the form $a(t-b)^{2}+c$ .

A particle moves along a straight line so that its velocity in $ \mathrm{m} \mathrm{s}^{-1} $, at time t seconds, is given by g(t) .   
2. Find the distance the particle travels between t=7 and t=10 .   

The function f is defin3fek+e e *vkvv(yk ;t+ed by


1. Determine whether or not f is continuous at x=1 .

The graph of the function g is obtained by applying the following transformations to the graph of f :
a horizontal translation 2 units to the left, followed by a reflection in the x -axis, followed by
a vertical stretch by a factor of 3 .
2. Find g(x) .

  Let $f(x)=3 x^{2}+12 x+9$ , for $x \in \mathbb{R}$ .
1. For the graph of f , find:
1. the y -intercept; (a,b) a =    b =   
2. the x -intercepts.

The function f can be written in the form $f(x)=a(x-h)^{2}+k$ . B (a,b) a =    b =    C(c,d) c =    d =   
2. Find the values of a, h and k .a =    h =    k =   
3. For the graph of f , write down:
1. the coordinates of the vertex; (a,b) a =    b =   
2. the equation of the axis of symmetry.x =   

The graph of a function g is obtained from the graph of f by a reflection in the x -axis, followed by a translation by the vector $ \binom{0}{4}$ .
4. Find g(x) , giving your answer in the form $g(x)=p x^{2}+q x+r$ .   

  Let $f(x)=2 x^{2}+4 x+p$ , for $x \in \mathbb{R}$ , where $p \in \mathbb{Z}$ .
1. The equation f(x)=0 has two equal roots.
1. Write down the value of the discriminant of f .   
2. Show that p=2 .
2. For the graph of f , find:
1. the equation of the axis of symmetry. x =   
2. the coordinates of the vertex; (a,b) a =    b =   
3. Write down the solution to the equation f(x)=0 . x =   
4. The function f can be written in the form $f(x)=a(x-h)^{2}+k$ . Find the values of a, h and k . a =    h =    k =   
5. The graph of a function g is obtained from the graph of f by a reflection in the x -axis. Find the coordinates of the vertex of the graph of g . (a,b) a =    b =   

  Let $f(x)=2 x^{2}-8 x+6$ , for $ x \in \mathbb{R}$ .
1. Write down the value of f(0) .   
2. Solve the equation f(x)=0 .

The function f can be written in the form $f(x)=a(x-h)^{2}+k$ .      
3. Find the values of a, h and k .a =    h =    k =   
4. For the graph of f , write down:
1. the coordinates of the vertex;(a,b) a =    b =   
2. the equation of the axis of symmetry.

The graph of a function g is obtained from the graph of f by a reflection in the x -axis, followed by a translation by the vector $\binom{1}{3}$ . x =   
5. Find g(x) , giving your answer in the form $g(x)=p x^{2}+q x+r $.   

$\text { Let } f(x)=a(x+1)(x+5) \text {, for } x \in \mathbb{R} \text {, where } a \in \mathbb{Z} \text {. The following diagram shows part of the graph of } f \text {. }$



The graph of f has x -intercepts at (p, 0) and (q, 0) , and a y -intercept at (0,-10) .
1. 1. Write down the value of p and the value of q .
2. Find the value of a .
2. Find the equation of the axis of symmetry.
3 . Find the coordinates of the vertex.

The graph of a function g is obtained from the graph of f by a reflection in the y -axis, followed by a translation by the vector $\binom{0}{2}$ . The point $\mathrm{P}(-2,6)$ on the graph of f is mapped to point $\mathrm{Q}$ on the graph of g .
4. Find the coordinates of Q .

Let $f(x)=2(x-1)^{2}-8$ , for $ x \in \mathbb{R} $.
1. Show that $f(x)=2 x^{2}-4 x-6 $.
2. For the graph of f :
1. write down the coordinates of the vertex;
2. write down the y -intercept;
3. find both x -intercepts.
3. Hence sketch the graph of f .

Let $g(x)=6 x^{2}$ , for $x \in \mathbb{R}$ .
The graph of f may be obtained from the graph of g using the following two transformations:
a compression of scale factor a in the y -direction, followed by
a translation of $ \binom{h}{k}$ .
4. Find the values of a, h and k .

The following diagram shows the gra*h*tec+g x*pbvuq nud k3/ v*2ph of y=f(x) . The graph has a horizontal asymptote at y=-2 . The graph crosses the x -axis at x=-2 kuvd/+t*c qbx2*vuh**gnp 3e and x=2 , and the y axis at y=2 .


On the following set of axes, sketch the graph of $y=[f(x)]^{2}-1$ , clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima.

The following diagram shows the graph of y=f(x) . Ttre-h)46u n/ru( mhruhz ..j ehe graph has a horizontal asymptote at y=-2 . The t(h.hn)6r/rh. emuez uu4j-r graph crosses the x -axis at x=-1 and x=1 , and the y axis at y=2 .


On the following set of axes, sketch the graph of $y=[f(x)]^{2}-2$ , clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima.

Consider the functions f(x):*zbk hqq3iirst arp(mq j*0) i87v.v =3 $\cos (x)+\frac{9}{2}$ and $ g(x)=3 \cos \left(x+\frac{\pi}{3}\right)+A$ , where $x \in \mathbb{R}$ and $A<\frac{9}{2}$ .
1. Describe a sequence of two transformations that transforms the graph of f to the graph of g .

The y -intercept of the graph g is at the point $\left(0, \frac{9}{2}\right)$
2. Find the range of q .

  $\text { Let } f(x)=a x^{3}+b x^{2}+c x+d \text {, for } x \in \mathbb{R} \text {, where } a, b, c, d \in \mathbb{Q} \text {. The diagram below shows part of the graph of } y=f(x) \text {. }$



1. Using the information shown in the diagram, find the values of a, b, c and d . a =    b =    c =    d =   
2. Let $g(x)=-\frac{3}{4} f(-2 x+1)$ . (x,y) x =    y =    (x,y) x =    y =    (x,y) x =    y =   
1. Find the coordinates of the points where the graph of y=g(x) intercepts the x -axis.
2. Find the y -intercept of the graph of y=g(x) .(x,y) x =    y =   

Consider the functiom.bg;b/kk h ;q d8hjm3to rq-4n $g(x)=a x^{3}+b x^{2}+c x+d$ , where $ x \in \mathbb{R}$ and a, b, c, d $\in \mathbb{R}$ .
1. 1. Write down an expression for $g^{\prime}(x)$ .
2. Hence, given that $ g^{-1}$ does not exist, show that $b^{2}-3$ a c>0 .

Consider the function $f(x)=\frac{x^{3}}{2}+3 x^{2}+6 x+\frac{9}{2}$
2. 1. Show that $f^{-1}$ exists.
2. f(x) can be written in the form $p(x+2)^{3}+q $, where p,$ q \in \mathbb{R}$ . Find the value of p and the value of q .
3. Hence, find $f^{-1}(x) $.

The graph of f(x) may be obtained by transforming the graph of y=x^{3} using a sequence of three transformations.
3. State each of the transformations in the order in which they are applied.
4. Sketch the graphs of y=f(x) and y=f^{-1}(x) on the same set of axes, indicating the points where each graph crosses the coordinate axes.

The following diagram shows thgtz ;710yk5qk b,gf ege graph of $y=\arctan (2 x-3)+\frac{3 \pi}{4} $ for $x \in \mathbb{R}$ , with asymptotes at $y=\frac{\pi}{4}$ and $y=\frac{5 \pi}{4}$ .



1. Describe a sequence of transformations that transforms the graph of $y=\arctan x$ to the graph of $y=\arctan (2 x-3)+\frac{3 \pi}{4}$ for $x \in \mathbb{R}$ .
2. Show that $\arctan p-\arctan q \equiv \arctan \left(\frac{p-q}{1+p q}\right) $.
3. Verify that $\arctan (x+2)-\arctan (x+1)=\arctan \left(\frac{1}{(x+1)^{2}+(x+1)+1}\right)$ .
4. Using mathematical induction and the results from part (b) and (c), prove that

$\sum_{r=1}^{n} \arctan \left(\frac{1}{r^{2}+r+1}\right)=\arctan (n+1)-\frac{\pi}{4} \quad \text { for } n \in \mathbb{Z}^{+}$