IGCSE Maths / Graphs and transformations
Graphs with reciprocal terms revision notes
Edexcel International GCSE Maths (4MA1), chapter 26: Graphs and transformations, specification point 3.3a. Free lesson.
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A reciprocal graph has x underneath a number, so its curve forms two separate branches. The position of the branches depends on whether the denominator is x or x².
How to find points on y = 6/x
To plot a reciprocal graph, substitute each given x-value into the equation and divide. For y = 6/x, a larger positive x gives a smaller positive y. A negative x gives a negative y, so those points lie in the opposite corner of the grid. When x = 1, y = 6/1 = 6. When x = 2, y = 6/2 = 3. When x = 3, y = 6/3 = 2. When x = −2, y = 6/(−2) = −3.
How to draw the two branches of y = 6/x
Plot the points, then draw two smooth, separate curves: one in the top-right corner and one in the bottom-left. As x gets larger, 6/x gets closer to 0, so the right-hand branch flattens towards the x-axis. As x gets closer to 0, the values grow in size, so the branches climb alongside the y-axis. The curve approaches the x-axis and y-axis but never touches either. An asymptote is a line that a curve gets closer to without reaching.
Why the graph has a gap at x = 0
Putting x = 0 into y = 6/x would mean dividing 6 by 0. This has no value, so there is no point on the graph at x = 0 and it cannot cross the y-axis. Leave a clear gap between the branches. The branches approach the y-axis from either side, but neither reaches it. Draw each branch on its own. Joining them through the middle would show a point where the equation has no value.
How x² changes the branches
For y = 8/x², squaring the denominator removes the sign of x. The denominator is positive for positive and negative x, so y is positive on both sides. The branches are therefore both above the x-axis, with matching heights for x-values that differ only by their sign. When x = 1, y = 8, giving the point (1, 8). When x = 2, y = 2, giving the point (2, 2). When x = −2, y = 2, giving the point (−2, 2). With one x underneath, the branches are in opposite corners. With x² underneath, both branches are above the x-axis.
Working out a value for a negative x
Substitute the x-value into the fraction and check the sign. A positive number divided by a negative number gives a negative answer.