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					OLD IB Problems with Quadratics for Review
1.    The diagram shows the graph of the function y = ax2 + bx + c.
                                               y

                                                                                  x




      Complete the table below to show whether each expression is positive, negative or zero.

            Expression              positive            negative               zero
            a
            c
            b2 – 4ac
            b


                                                                                                      (Total 4 marks)

2.   A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the
ground after t seconds is given by

                                                h = 2 + 20t – 5t2, t  0

      (a)       Find the initial height above the ground of the ball (that is, its height at the instant when
                it is released).
                                                                                                                   (2

      (b)       Show that the height of the ball after one second is 17 metres.
                                                                                                                  (2)

      (c)       At a later time the ball is again at a height of 17 metres.

                (i)    Write down an equation that t must satisfy when the ball is at a height of 17 metres.

                (ii)   Solve the equation algebraically.
                                                                                                                  (4)


                                                                                                      (Total 8 marks)




                                                                                                                   1
3.   Consider the function f (x) = 2x2 – 8x + 5.

     (a)   Express f (x) in the form a (x – p)2 + q, where a, p, q      , by completing the square.

     (b)   Find the minimum value of f (x).

                                                                                                 (Total 6 marks)

4.   The equation kx2 + 3x + 1 = 0 has exactly one solution. Find the value of k.

                                                                                                 (Total 6 marks)

5.   The equation x2 – 2kx + 1 = 0 has two distinct real roots. Find the set of all possible values of k.

                                                                                                 (Total 6 marks)

6.   The equation of a curve may be written in the form y = a(x – p)(x – q). The curve intersects the
     x-axis at A(–2, 0) and B(4, 0). The curve of y = f (x) is shown in the diagram below.

                                                   y
                                                   4


                                                   2

                                           A                     B
                                 –4       –2     0         2      4        6 x

                                                –2


                                                –4


                                                –6



     (a)   (i)     Write down the value of p and of q.

           (ii)    Given that the point (6, 8) is on the curve, find the value of a.

           (iii)   Write the equation of the curve in the form y = ax2 + bx + c.
                                                                                                             (5)

7.   Part of the graph of f (x) = (x – p) (x – q) is shown below.




     The vertex is at C. The graph crosses the y-axis at B.
     (a) Write down the value of p and of q.
     (b) Find the coordinates of C.
     (c) Write down the y-coordinate of B.



                                                                                                              2
8.   The following diagram shows part of the graph of a quadratic function, with equation in the
     form y = (x − p)(x − q), where p, q  .




     (a)   Write down

           (i)    the value of p and of q;

           (ii)   the equation of the axis of symmetry of the curve.
                                                                                                          (3)




     (b)   Find the equation of the function in the form y = (x − h)2 + k, where h, k    .
                                                                                                          (3)



                                                                                              (Total 6 marks)




                                                                                                           3

				
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