Graphical Solutions

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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.

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.

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.

Let us summarize the number of intersections in the following table then determine the highest value of k where there are three intersections.

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.