Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

EXERCISE 2E Sketch the following curves, marking clearly the values of x and y where they

cross the co-ordinate axes.

1 y = x(x − 3)(x + 4) 2 y = (x + 1)(2x − 5)(x − 4)

3 y = (5 − x)(x − 1)(x + 3) 4 y = x^2 (x − 3)

5 y = (x + 1)^2 (2 − x) 6 y = (3x − 4)(4x − 3)^2

7 y = (x + 2)^2 (x − 4)^2 8 y = (x − 3)^2 (4 + x)^2

9 Suggest an equation for this curve.

●?^ What happens to the curve of a polynomial if it has a factor of the form

(x − a)^3? Or (x − a)^4?

Curves of the form y = (^) x—^1 n (for x ≠ 0)
The curves for n = 3, 5, ... are not unlike that for n = 1, those for n = 4, 6, ... are
like that for n = 2. In all cases the point x = 0 is excluded because^10 is undefined.
x
y
–2 –1 0 1 2 3
4
x
y
y =
O
x
y
O
(^1) x 2

-^1 x

y =

x

y

y =

O

x

y

O

(^1) x 2

-^1 x

y =

Figure 2.32

(a) n = 1, y = (^1) x (b) n = 2, y =
1
x^2