4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY. Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 20.
(i) Sketch the graph of $y = \sqrt { x } - \frac { x } { 4 }$ for $x \geqslant 0$, and find the coordinates of the turning point.
(ii) Describe in words how the graph of $y = \sqrt { 4 x + 1 } - x - 1$ for $x \geqslant - \frac { 1 } { 4 }$ is related to the graph that you sketched in part (i). Write down the coordinates of the turning point of this new graph.
Point $A$ is at $( - 1,0 )$ and point $B$ is at $( 1,0 )$. Curve $C$ is defined to be all points $P$ that satisfy the equation $| A P | \times | B P | = 1$, that is; the distance from $P$ to $A$, multiplied by the distance from $P$ to $B$, is 1 .
(iii) Find all points that lie on both the $x$-axis and also on the curve $C$.
(iv) Find an equation in the form $y = f ( x )$ for the part of the curve $C$ in the region where $x > 0$ and $y > 0$. You should explicitly determine the function $f ( x )$.
(v) Use part (ii) to determine the coordinates of any turning points of the curve $C$ in the region where $x > 0$ and $y > 0$.
(vi) Sketch the curve $C$, including any parts of the curve with $x < 0$ or $y < 0$ or both.
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