DP1_Date_4 October 2024_Worksheet
Functions [39 marks]
1. [Maximum mark: 5]
The quadratic equation (k − 1)x2 + 2x + (2k − 3) = 0, where
k ∈ R, has real distinct roots.
Find the range of possible values for k. [5]
2. [Maximum mark: 16]
(a) The graph of a quadratic function f has its vertex at the point
(3, 2) and it intersects the x-axis at x = 5. Find f in the form
2
f (x) = a(x − h) + k.
[3]
The quadratic function g is defined by g(x) 2
= px + (t − 1)x − p where
x ∈ R and p, t ∈ R, p ≠ 0.
In the case where g(−3) = g(1) = 4,
(b.i) find the value of p and the value of t. [4]
(b.ii) find the range of g. [3]
(c) The linear function j is defined by j(x) = −x + 3p where
x ∈ R and p ∈ R, p ≠ 0.
Show that the graphs of j(x) = −x + 3p and
+ (t − 1)x − p have two distinct points of
2
g(x) = px
intersection for every possible value of p and t. [6]
3. [Maximum mark: 4]
Consider the equation kx2 − (k + 3)x + 2k + 9 = 0, where k ∈ R.
(a) Write down an expression for the product of the roots, in terms
of k. [1]
(b) Hence or otherwise, determine the values of k such that the
equation has one positive and one negative real root. [3]
4. [Maximum mark: 7]
Consider the function f (x) = −2(x − 1)(x + 3), for x ∈ R. The
following diagram shows part of the graph of f .
For the graph of f
(a.i) find the x-coordinates of the x-intercepts. [2]
(a.ii) find the coordinates of the vertex. [3]
(b) The function f can be written in the form
2
f (x) = −2(x − h) + k.
Write down the value of h and the value of k. [2]
5. [Maximum mark: 7]
The equation 3px2 + 2px + 1 = p has two real, distinct roots.
(a) Find the possible values for p. [5]
(b) Consider the case when p = 4. The roots of the equation can
a±√ 13
be expressed in the form x =
6
, where a ∈ Z. Find the
value of a. [2]
© International Baccalaureate Organization, 2024
