| Date | November 2012 | Marks available | 4 | Reference code | 12N.1.hl.TZ0.3 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Find | Question number | 3 | Adapted from | N/A |
Question
Let \(f(x) = {x^3} + a{x^2} + bx + c\) , where a , b , \(c \in \mathbb{Z}\) . The diagram shows the graph of y = f(x) .
Using the information shown in the diagram, find the values of a , b and c .
If g(x) = 3f(x − 2) ,
(i) state the coordinates of the points where the graph of g intercepts the x-axis.
(ii) Find the y-intercept of the graph of g .
Markscheme
METHOD 1
f(x) = (x + 1)(x − 1)(x − 2) M1
\( = {x^3} - 2{x^2} - x + 2\) A1A1A1
a = −2 , b = −1 and c = 2
METHOD 2
from the graph or using f(0) = 2
c = 2 A1
setting up linear equations using f(1) = 0 and f(–1) = 0 (or f(2) = 0) M1
obtain a = −2 , b = −1 A1A1
[4 marks]
(i) (1, 0) , (3, 0) and (4, 0) A1
(ii) g(0) occurs at 3f(−2) (M1)
= −36 A1
[3 marks]
Examiners report
This question was well answered in general. Part b(ii) was often the most problematic, usually because of candidates going to the trouble of finding an explicit and sometimes incorrect expression for f(x − 2).
This question was well answered in general. Part b(ii) was often the most problematic, usually because of candidates going to the trouble of finding an explicit and sometimes incorrect expression for f(x − 2).
