WORKSHEET QUADRATICS
1. Consider f(x) = 2kx2 – 4kx + 1, for k ≠ 0. The equation f(x) = 0 has two equal roots.
(a) Find the value of k.
(b) The line y = p intersects the graph of f. Find all possible values of p.
2. The following diagram shows part of the graph of a quadratic function f.
The xintercepts are at (–4, 0) and (6, 0) and the yintercept is at (0, 240).
(a) Write down f(x) in the form f(x) = –10(x – p)(x – q).
(b) Find another expression for f(x) in the form f(x) = –10(x – h)2 + k.
(c) Show that f(x) can also be written in the form f(x) = 240 + 20x – 10x2.
A particle moves along a straight line so that its velocity, v m s–1, at time t seconds is given by
v = 240 + 20t – 10t2, for 0 ≤ t ≤ 6.
(d) (i) Find the value of t when the speed of the particle is greatest.
(ii) Find the acceleration of the particle when its speed is zero.
3. Let f(x) = 8x – 2x2. Part of the graph of f is shown below.
(a) Find the xintercepts of the graph.
(b) (i) Write down the equation of the axis of symmetry.
(ii) Find the ycoordinate of the vertex.
4. Let f(x) = p(x – q)(x – r). Part of the graph of f is shown below.
The graph passes through the points (–2, 0), (0, –4) and (4, 0).
(a) Write down the value of q and of r.
(b) Write down the equation of the axis of symmetry.
(c) Find the value of p.
5. The quadratic equation kx2 + (k – 3)x + 1 = 0 has two equal real roots.
(a) Find the possible values of k.
(b) Write down the values of k for which x2 + (k – 3)x + k = 0 has two equal real roots.
6. Let f (x) = 3(x + 1)2 – 12.
(a) Show that f (x) = 3x2 + 6x – 9.
(b) For the graph of f
(i) write down the coordinates of the vertex;
(ii) write down the equation of the axis of symmetry;
(iii) write down the yintercept;
(iv) find both xintercepts.
(c) Hence sketch the graph of f.
7. Let f (x) = 2x2 – 12x + 5.
(a) Express f(x) in the form f(x) = 2(x – h)2 – k.
(b) Write down the vertex of the graph of f.
(c) Write down the equation of the axis of symmetry of the graph of f.
(d) Find the yintercept of the graph of f.
(e) The xintercepts of f can be written as , where p, q, r .
Find the value of p, of q, and of r.
8. Part of the graph of the function y = d (x −m)2 + p is given in the diagram below.
The xintercepts are (1, 0) and (5, 0). The vertex is V(m, 2).
(a) Write down the value of
(i) m;
(ii) p.
(b) Find d.
9. Consider the function f (x) = 2x2 – 8x + 5.
(a) Express f (x) in the form a (x – p)2 + q, where a, p, q Î .
(b) Find the minimum value of f (x).
10. The diagram shows the graph of the function y = ax2 + bx + c.
Complete the table below to show whether each expression is positive, negative or zero.
Expression

positive

negative

zero

a




c




b2 – 4ac




b




11. The diagram shows the parabola y = (7 – x)(l + x). The points A and C are the xintercepts and the point B is the maximum point.