(a) The grid shows the graph of \(y=a+b x^{2}\).
The graph passes through the points with coordinates \((0,4)\) and \((1,1)\).

(i) Find the value of \(a\) and the value of \(b\).
\[
\begin{aligned}
&a= \\
&b=
\end{aligned}
\] ............................................ .. [2]
............................................

(ii) Write down the equation of the tangent to the graph at \((0,4)\).
[1]

(iii) The equation of the tangent to the graph at \(x=-1\) is \(y=6 x+7\).
Find the equation of the tangent to the graph at \(x=1\).
[2]

(b) The table shows some values for \(y=1+\frac{5}{3-x}\) for \(-2 \leqslant x \leqslant 1.5\).

(i) Complete the table. [3]

(ii) On the grid, draw the graph of \(y=1+\frac{5}{3-x}\) for \(-2 \leqslant x \leqslant 1.5\). [4]

(c) (i) Write down the values of \(x\) where the two graphs intersect.
\[
x=\ldots \ldots \ldots \ldots \ldots \text { or } x=
\] .............. [2]

(ii) The answers to part(c)(i) are two solutions of a cubic equation in terms of \(x\).
Find this equation in the form \(a x^{3}+b x^{2}+c x+d=0\), where \(a, b, c\) and \(d\) are integers.
[4]

The table shows some values for $$\(y=1+\frac{5}{3-x}\)$$ for $$\(-2 \leqslant x \leqslant 1.5\)$$. Complete the table.

Answer:

Knowledge points:

Solution:

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