Figure 1 shows a sketch of the curve with equation $y = \frac{3}{x}, \quad x \neq 0.$ (a) On a separate diagram, sketch the curve with equation $y = \frac{3}{x+2}, \quad x \neq -2,$ showing the coordinates of any point at which the curve crosses a coordinate axis - Edexcel - A-Level Maths Pure - Question 7 - 2007 - Paper 1

Identify the Characteristics of the Curve:
The given function is similar to the original function $y = \frac{3}{x}$, but is translated horizontally.

Finding the Asymptotes:
The vertical asymptote occurs where the denominator is zero. Therefore, for the equation $y = \frac{3}{x + 2}$, the vertical asymptote is at
[ x + 2 = 0 \implies x = -2. ] The horizontal asymptote is $y = 0$, as the function approaches zero as $x$ approaches infinity or negative infinity.

Intercepts:
To find the x-intercept, set $y = 0$:
[ 0 = \frac{3}{x + 2} ]
which does not yield any valid $x$. Therefore, the curve does not intersect the x-axis.
To find the y-intercept, set $x = 0$:
[ y = \frac{3}{0 + 2} = \frac{3}{2}. ]
Thus, the curve crosses the y-axis at the point ((0, \frac{3}{2})).

Sketch:
On a new diagram, sketch the curve, showing the asymptotes and the point of intersection at the y-axis.