7 A graph has equation
$$y = \frac { x - 4 } { x ^ { 2 } + 9 }$$
- Explain why the graph has no vertical asymptote and give the equation of the horizontal asymptote.
- Show that, if the line \(y = k\) intersects the graph, the \(x\)-coordinates of the points of intersection of the line with the graph must satisfy the equation
$$k x ^ { 2 } - x + ( 9 k + 4 ) = 0$$
- Show that this equation has real roots if \(- \frac { 1 } { 2 } \leqslant k \leqslant \frac { 1 } { 18 }\).
- Hence find the coordinates of the two stationary points on the curve.
(No credit will be given for methods involving differentiation.)