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Graphical Solutions

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IGCSE,mathematics,past papers,revisions,mark scheme,CIE,examinations,graphs,lines,gradients,tangents,intersection,curves
This item is taken from IGCSE Mathematics (0580) Paper 43 of May/June 2011.


To deal with this item, let us summarize the details of the problem:
   (a) Syllabus area: Algebra
   (b) Specific topic: Graphical Solutions



For (i), we should take note that y = f(x).
Since the given is f(2), then x = 2. This suggest that we should draw the line x = 2 then find the value of y in the intersection of the line x = 2 and the given graph.
IGCSE,mathematics,past papers,revisions,mark scheme,CIE,examinations,graphs,lines,gradients,tangents,intersection,curves
For (ii), since y = f(x) and the given is f(x) = 0, then y = 0. This means that the line y = 0 must be drawn. The line y = 0 is also the x-axis. Find the value of x in the intersection between the x-axis and the given graph.
IGCSE,mathematics,past papers,revisions,mark scheme,CIE,examinations,graphs,lines,gradients,tangents,intersection,curves
For (iii), we are looking for the value of k where there are three solutions. This means that we are looking for a horizontal line that passes through the given graph at exactly three points. We should also take note that the value of k is an integer. The possible values of k are 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 (based on the range of the grid given). Let us draw the horizontal lines and mark the intersection(s) of each.
IGCSE,mathematics,past papers,revisions,mark scheme,CIE,examinations,graphs,lines,gradients,tangents,intersection,curves
Let us summarize the number of intersections in the following table then determine the highest value of k where there are three intersections.
IGCSE,mathematics,past papers,revisions,mark scheme,CIE,examinations,graphs,lines,gradients,tangents,intersection,curves
For (iv), to solve for the equation f(x) = x, we should take note that f(x) = y. Hence, f(x) = x is the same as y = x. By now, you should be familiar with the graph of this equation. If not, you may use a table to assign values of x then get the corresponding values of y. The graph is a straight line passing through the origin and slanting to the right.
IGCSE,mathematics,past papers,revisions,mark scheme,CIE,examinations,graphs,lines,gradients,tangents,intersection,curves

Let us determine the coordinates of the intersection between the line y=x and the given graph.
IGCSE,mathematics,past papers,revisions,mark scheme,CIE,examinations,graphs,lines,gradients,tangents,intersection,curves
Therefore,
IGCSE,mathematics,past papers,revisions,mark scheme,CIE,examinations,graphs,lines,gradients,tangents,intersection,curves
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