The graph shows the relation of the form \(y = mx^2 + nx + r\), where m, n and r are constants. Using the graph: (a) state the scale used on both axes; (b) ...
Question 1 Report
The graph shows the relation of the form \(y = mx^2 + nx + r\), where m, n and r are constants.
Using the graph:
(a) state the scale used on both axes; (b) find the values of m, n and r; (c) find the gradient of the line through P and Q; (d) state the range of values of x for which y > Q.
Answer Details
(a) Scale
- x-axis: 2 cm represents 2 units, i.e. 1 cm represents 1 unit.
- y-axis: 2 cm represents 10 units, i.e. 1 cm represents 5 units.
(b) Values of \(m\), \(n\) and \(r\)
From the graph, the curve cuts the \(x\)-axis at \(x=-2\) and \(x=4\). Hence,
\[y=-(x+2)(x-4)\]
\[y=-\left(x^2-2x-8\right)=-x^2+2x+8.\]
Comparing with \(y=mx^2+nx+r\),
\[\boxed{m=-1,\quad n=2,\quad r=8.}\]
(c) Gradient of \(PQ\)
From the graph, \(P(-5,-27)\) and \(Q(3,5)\).
\[\text{Gradient of }PQ=\frac{5-(-27)}{3-(-5)}=\frac{32}{8}=\boxed{4}.\]
(d) Range of \(x\) for which \(y>0\)
The graph is above the \(x\)-axis between its roots \(-2\) and \(4\). Therefore,
\[\boxed{-2<x<4}.\]
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