Let $f(x)=x^{2}-2 x-8$ , for x $\in \mathbb{R}$ .
1. Write down the y -intercept of the graph of y=f(x) .   
2. Solve the equation f(x)=0 .      
3. Find the equation of the axis of symmetry of the graph of y=f(x) .  (代数式) 

  The following diagram shows part of the graph *p 5yie,.))q xedho kox 9ipnc* s-d) uhzul;;of a quadratic function f.


The vertex is at point $ \mathrm{A}$ and the x -intercepts are at points $\mathrm{B} $ and $\mathrm{C}$ . The function f can be written in the form $ f(x)=(x-h)^{2}+k$ .
1. Write down the values of h and k .

The function f can also be written in the form f(x)=(x+p)(x-q) .h =    k =   
2. Write down the values of p and q . p =    q =   
3. Find the y -intercept of the graph of f .   

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

Let $f(x)=x^{2}-4 x+3$ , for x $\in \mathbb{R}$ .
1. For the graph of y=f(x) , find:
1. the y -intercept;
2. the x -intercepts.
2. The function f can be expressed in the form $f(x)=(x-h)^{2}+k$ . Find the value of h and the value of k .
3. Sketch the graph of y=f(x) on the grid below. Clearly label the intercepts with the axes, and the vertex.

  The equation $x^{2}+k x+4=0$ has two equal roots.
Find the possible values of k . ±   

  Let $f(x)=a(x-h)^{2}+k$ , for $x \in \mathbb{R}$ .
The vertex of the graph of f is at $\mathrm{P}(3,4)$ and the graph passes through $\mathrm{Q}(1,-4)$ .
1. Write down the values of h and k . h =    k =   
2. Find the value of a .   

  The diagram below shows the graph of *u fbx7 l6+qoma quadratic function $f(x)=2 x^{2}+b x+c $.




1. Write down the value of c .   
2. Find the value of b and write down f(x) .  (代数式) 
3. Calculate the coordinates of the vertex of the graph of f .(a,b) a =    b =   

The equation $x^{2}-k x+1=0$ has two distinct real roots.
Find the possible values of k .

  1. Express $2 x^{2}-8 x+9$ in the form a(x+b)^{2}+c where a, b, $c \in \mathbb{Z}$ .   
2. Given that f(x)=x-2 and $(g \circ f)(x)=2 x^{2}-8 x+9$, find g(x) .   

  Consider $f(x)=x^{2}+b x+c$ , for x $\in \mathbb{R}$ , where b, c $\in \mathbb{Z}$ . The graph of f has a local minimum at x=2.5 . The distance between the two x -intercepts of the graph of f is 7 .
1. Find the coordinates of the two x -intercepts. (a,b) a =    b =    (c,d) c =    d =   
2. Find the value of b and the value of c .b =    c =   

  Let $f(x)=a x^{2}-24 x+c$ , for $x \in \mathbb{R}$ , where a, c $\in \mathbb{Z}$ .
1. A horizontal line, L , intersects the graph of y=f(x) at x=1 and x=7 .
1. Find the equation of the axis of symmetry of the graph of y=f(x) .   
2. Hence show that a=3 .   
2. The equation of L is y=6 . Find the value of c .   

  The graph shows the curve of a quadratic function of the for3g- l v41 ix+c(v-nim8tftere m $ f(x)=a x^{2}+b x+90 $.



1. Write down the equation for the axis of symmetry of the curve.   
2. Hence, or otherwise, find the value of a and the value of b . a =    b =   
3. Find the y -coordinate of the vertex of the curve.   

The equation $x^{2}+(k-3) x-3$ k=0 has two distinct real solutions.
Find the possible values of k .

  1. Express $3 x^{2}+18 x+20$ in the form $a(x+b)^{2}+c$ where a, b, c $\in \mathbb{Z}$ .   
2. Given that f(x)=x+3 and $(g \circ f)(x)=3 x^{2}+18 x+20$ , find g(x) .   

  Let $f(x)=-6 x^{2}+b x+c$ , for x $\in \mathbb{R}$ , where b, c $\in \mathbb{Z}$ .
1. A horizontal line, L , intersects the graph of y=f(x) at (-1, h) and (5, h) , where $h \in \mathbb{R}$ .
1. Find the equation of the axis of symmetry of the graph of y=f(x) .   
2. Hence show that b=24 .   
2. Given h=-8 , find the value of c .   

  A farmer is going toy78wz - c) aes8zantr; fence two equal adjacent parcels of land. These parcels share one side (which also requi8z r taw;secn)-z78ayres fencing) as shown in the following diagram. The farmer has only
80 metres of fence.


1. Write down the equation for the total length of the fence, 80 $\mathrm{~m}$ , in terms of x and y . y =  (代数式) 
2. Write down the total area of both parcels of land in terms of x .  (代数式) 
3. Find the maximum area, in $\mathrm{m}^{2}$ , of one parcel of land. ≈   

  The graph of a quadrqlzk8xs i/q+y;/k:v z atic function has a y -intercept at $\mathrm{A}(0,24)$ and one of its x -intercepts is $\mathrm{B}(2,0)$ .
The x -coordinate of the vertex of the graph is 4 .
The equation of the quadratic function is in the form y=a $x^{2}+b x+c$ .
1. Write down the value of c .   
2. Find the value of a and the value of b .a =    b =   
3. Write down the coordinates of the second x -intercept of the function. (a,b) a =    b =   

Let $f(x)=x^{2}+k x$ and g(x)=x+k , for $x \in \mathbb{R}$ , where k is a constant.
The graphs of y=f(x) and y=g(x) intersect at two distinct points.
Find the possible values of k .

  The line $ y=x+1 $ touches the graph of the function $ g(x)=2 x^{2}+k x+3 $ at one point. Find the possible values of k .      

Let $f(x)=p-\frac{16}{x}$ , for $x \neq 0$ , where p is a constant.
The line y=x-p intersects the graph of y=f(x) at two distinct points.
Find the possible values of p .

  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. (a,b) a =    b =    (c,d) c =    d =   

The function f can be written in the form $f(x)=a(x-h)^{2}+k$ .
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.

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 $g(x)=p x^{2}+q x+r$ .  (代数式) 

  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.   
2. the coordinates of the vertex; (a,b) a =    b =   
3. Write down the solution to the equation f(x)=0 .   
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 $\binom{1}{3}$ .   
5. Find g(x) , giving your answer in the form $g(x)=p x^{2}+q x+r$ .  (代数式) 

Let f(x)=a(x+1)(x+5) , neds+syw9wrx-0l 3 9rfor $ x \in \mathbb{R} $, where $ a \in \mathbb{Z}$ . The following diagram shows part of the graph of f .



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 .

  The quadratic equati9( bu7w6y+cnw 3nj09qsuod+v p1ukunon $ x^{2}-k x+(k-3)=0 $ has roots $ \alpha $ and$ \beta $ such that $\alpha^{2}+\beta^{2}=6$. Without solving the equation, find the possible values of the real number k .      

  $\text { The quadratic equation } x^{2}-k x+(k-1)=0 \text { has roots } \alpha \text { and } \beta \text {. Without solving the equation, find the possible values of the real number } k \text { given that } \alpha^{2}+\beta^{2}=17 \text {. }$      

  A cannonball is fired from the top of a tower. The rate of chanvq06a xk( kssg03vbt 2ge of the height, h , of the cannonball abk3 2s(00 gkxsqt6vb avove the ground is modelled by

$h^{\prime}(t)=-4 t+20, t \geq 0$,

where h is in metres and t is the time, in seconds, since the moment the cannonball was fired.
1. Determine the time t at which the cannonball reached its maximum height.

After one second, the cannonball is 26 metres above the ground.   
2. 1. Find an expression for h(t) , the height of the cannonball above the ground at time t .  (代数式) 
2. Hence, determine the maximum height reached by the cannonball.   
3. Write down the height of the tower.   
4. Calculate the height of the cannonball 4 seconds after it was fired.

The cannonball hits its target on the ground n seconds after it was fired.   
5. Find the value of n .≈   
6. Determine the total time the cannonball was above the height of the tower.

A second cannonball is fired from exactly halfway up the tower, with the same projectile motion as the first cannonball.   
7. Given that both cannonballs land at the same time, determine the length of time between the first cannonball and the second cannonball being fired.   

The axis of symmetry of the graph of a quadratic furm3w gyohb1y r/ 8u(hi,9+ azlnction has equation $x=\frac{3}{2}$ .
1. Draw the axis of symmetry on the grid below.

The graph of the quadratic function intersects the x -axis at the point $\mathrm{P}$(-1,0) . There is a second point, $\mathrm{Q}$ , at which the graph of the quadratic function intersects the x -axis.
2. Mark and label P and Q on the grid above.

The graph of the quadratic function has equation $y=-x^{2}+b x+c$ , where b, c $\in \mathbb{Z}$ .
3. 1. Find the value of b and the value of c .
2. Find the coordinates of the vertex, M.
3. Draw the graph of the quadratic function on the grid above.

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 .

  Consider $ f(x)=\log _{k}\left(8 x-2 x^{2}\right)$ , for 00 . The equation f(x)=3 has exactly one solution. Find the value of k .   

  When $p(x)=x^{2}-2 x+p$ is divided by x-r , the remainder is 4 .
Given that p, r $\in \mathbb{R}$ , find the largest possible value for p .   

  Let $f(x)=16-x^{2}$ , for x $\in \mathbb{R}$ .
1. Find the x -intercepts of the graph of f .

The following diagram shows part of the graph of f .

Rectangle $\mathrm{ABCD}$ is drawn with $\mathrm{A} \& \mathrm{~B}$ on the x -axis and $\mathrm{C} \& \mathrm{D}$ on the graph of f .
Let $\mathrm{OA}=a$ . P (a,b) a =    b =    Q (c,d) c =    d =   
2. Show that the area of $\mathrm{ABCD}$ is 32 a-2 $a^{3}$ .  (代数式) 
3. Hence find the value of a>0 such that the area of ABCD is a maximum.

Let $g(x)=(x-4)^{2}+k$ , for x $\in \mathbb{R}$ , where k is a constant. $\frac{a \sqrt{b}}{c}$ a =    b =    c =   
4. Show that when the graphs of f and g intersect, $2 x^{2}-8 x+k=0$ .
5. Given that the graphs of f and g intersect only once, find the value of k .   

  When $p(x)=x^{2}-2 x-ca$ is divided by (x-r) , the remainder is -5 . Given that c, r $\in \mathbb{R}$ , find the smallest possible value for c .   

  A quadratic functionrgc 4::y1aar u f can be written in the form f(x)=p(x-1)(x-q) .
The graph of y=f(x) has axis of symmetry x=2 and y -intercept at (0,-3) .
1. Find the value of q .   
2. Find the value of p .

The line y=k x+6 is a tangent to the graph of y=f(x) .   
3. Find the possible values of k .      

Consider the quadratic py8,neekwe / *nolynomial p(x)=2 k $x^{2}+15 k x+3 k+6$ , where k $\in \mathbb{R}$ .
1. Find an expression for the product of the roots of p(x) in terms of k .
2. Find the values of k for which p(x) has two distinct negative real roots.