2019 Nov P1 Q4

Question 4

Below are the graphs of $\: f(x)=x^2+bx-3$ and $\: g(x)=\frac{\displaystyle a}{\displaystyle x+p}$ . $f$ has a turning point at $C$ and passes through the $x$ -axis at $(1\;;\,0)$ . $D$ is the $y$ -intercept of both $f$ and $g$ . The graphs $f$ and $g$ also intersect each other at $E$ and $J$ . The vertical asymptote of $g$ passes through the $x$ -intercept of $f$ .

$4.1$	Write down the value of $p$ .	$(1)$
$4.2$	Show that $a=3$ and $b=2$ .	$(3)$
$4.3$	Calculate the coordinates of $C$ .	$(4)$
$4.4$	Write down the range of $f$ .	$(2)$
$4.5$	Determine the equation of the line through $C$ that makes an angle of $45^\circ$ with the positive $x$ -axis. Write your answer in the form $y=...$	$(3)$
$4.6$	Is the straight line, determined in QUESTION $4.5$ , a tangent to $f$ ? Explain your answer.	$(2)$
$4.7$	The function $h(x)=f(m-x)+q$ has only one $x$ -intercept at $x=0$ . Determine the values of $m$ and $q$ .	$(4)$
		$\textbf{[19]}$