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This page sets out the method for cubic and reciprocal graphs in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Cubic and reciprocal graphs topic page.
When you see in an equation and are asked to draw the graph.
When asked to match equations to sketches, use these cues.
On the curve y=x^3, the y-value when x=2 is 8.
A negative number cubed stays negative.
On the curve y=x^3+1, the y-value when x=2 is 9.
On the curve y=12/x, the y-value when x=3 is 4.
The vertical asymptote of y=1/x is x=0. The curve never crosses the y-axis.
A cubic keeps growing. A reciprocal changes sign across the asymptote.
y=x^3 passes through (-2, -8), (-1, -1), (0, 0), (1, 1) and (2, 8). The left end is down and the right end is up. y=-x^3 reflects that in the x-axis: the left end is up and the right end is down, through (2, -8). Neither curve turns back on itself the way a quadratic does. If your sketch comes back down after rising, you have drawn a quadratic by habit.
(-2)^3=(-2)×(-2)×(-2). The first product is +4, and 4×(-2)=-8. Writing +8 treats a cube as if it were a square. The sign of an odd power matches the sign of the input. That single fact fixes most cubic tables.
Method: 1. Plot y=x^3 through (-2, -8), (-1, -1), (0, 0), (1, 1) and (2, 8). 2. The left-hand end of this curve goes down and the right-hand end goes up. 3. For y=-x^3, reflect that shape in the x-axis so the curve passes through (2, -8).
y=12/x has no value at x=0, because division by zero is not defined. The curve splits into two branches. In the first quadrant it falls from a large positive value, through (1, 12), (2, 6), (3, 4) and (4, 3), getting closer to the x-axis. In the third quadrant the values are the negatives: (-1, -12), (-2, -6), and so on. You do not join the branches across the y-axis. There is a gap at x=0.
The curve does not touch either axis. A reciprocal that crosses an axis, or that is drawn as one connected wiggle, is a different graph. Points close to zero, such as x=0.5 if the table allows it, show the branch shooting upwards: 12/0.5=24. Those points are worth plotting when you are asked to sketch the steep part.
Method: 1. Do not plot y=12/x at x=0, because division by zero is not defined. 2. Draw the graph as two separate branches. 3. In the first quadrant the branch falls through (1, 12), (2, 6), (3, 4) and (4, 3).
A straight line is linear. A U is quadratic. A smooth curve with both ends going opposite ways, such as y=x^3, is cubic. Two branches in opposite corners are reciprocal. When you match an equation to a sketch, write one reason: "two branches, so reciprocal" or "both ends in opposite directions, so cubic". The reason protects the mark if the sketch is a bit ambiguous.
A point lies on the curve only if it satisfies the equation. For y=12/x, the point (2, 6) works because 12/2=6. The point (3, 5) does not, because 12/3=4, not 5. Substitute. Do not judge by eye once you have the equation.
When you match sketches in a list, go through them in the same order every time. A straight line is degree 1. A single smooth U, or an upside-down U, is degree 2. A curve that flattens near the origin and then runs away in opposite vertical directions is degree 3. Two separate branches are the reciprocal. Writing that one-line reason next to each choice is slower than guessing, and it is also how a wrong first impression gets corrected before you commit to it.
Method: 1. A straight graph is linear. 2. A smooth curve with its ends going in opposite directions, such as y=x^3, is a cubic. 3. Two branches sitting in opposite corners of the grid are a reciprocal graph.
Exam questions often ask about y = x^3 + c or y = x^3 - c rather than plain y = x^3. Adding or subtracting a number slides the whole curve up or down by that amount. The shape does not change. The curve y = x^3 - 4 is the curve y = x^3 moved down four units, so it crosses the y-axis at (0, -4).
A sketch needs the shape and one or two labelled points. Draw the left end going down and the right end going up for a positive x^3, and the opposite for y = -x^3. Label the y-intercept by putting x = 0. For y = x^3 - 8 the curve meets the x-axis where x^3 = 8, so at x = 2. A multiple choice question usually has several sketches with different intercepts, so the intercept decides which one is correct.
Method: 1. Decide the direction of the ends from the sign in front of x^3. 2. Put x = 0 to find the y-intercept and label it. 3. Put y = 0 to find where the curve crosses the x-axis, when that is easy. 4. Draw one smooth curve through the labelled points without kinks.
Tip: In a multiple choice sketch question, find the y-intercept first. It usually rules out two options straight away.
A common one mark question gives four or five equations and asks which is linear, quadratic, cubic or reciprocal. The highest power of x decides it. A plain x gives a straight line, x^2 gives a U or an upside-down U, x^3 gives a cubic, and 1/x or k/x gives a reciprocal curve with two branches. Say the word, and give the reason if the question asks for one.
You can also recognise a graph from a table. If the y values for x and -x are the same, the curve is a quadratic. If they are the same size but opposite in sign, the curve is a cubic y = x^3 or a reciprocal y = k/x. If the y values keep the same difference when x goes up by one, the graph is a straight line. Use this to spot a wrong entry in a table.
Method: 1. Find the highest power of x in the equation, or look for x in the denominator. 2. Name the type of graph and give one feature as a reason. 3. For a table, compare the y values at x and -x. 4. Mark any entry that breaks the pattern and recalculate it.
If a point lies on a curve, its coordinates satisfy the equation. Substitute them to find an unknown constant. For y = x^3 + a through (2, 15) put x = 2 and y = 15 to get 15 = 8 + a, so a = 7. For y = k/x through (4, 3) put in the values to get 3 = k/4, so k = 12. After finding the constant, write the full equation.
The same idea tests whether a point lies on a curve. Put the x value into the equation and see whether you get the stated y value. If you do, the point is on the curve. If you do not, it is not. Always show the substitution, because the answer yes or no is worth nothing without it. A line and a curve meet where their equations are equal, so set them equal and solve.
Method: 1. Write the equation with the unknown constant in it. 2. Substitute the x and y values from the given point. 3. Solve the equation for the unknown. 4. Write the final equation and test it with the point.
Inverse proportion produces a reciprocal curve. If the time to complete a job falls as more people work on it, the formula is T = k/n with k the total work. Doubling the number of people halves the time. Questions ask you to substitute, to find the number of people for a given time, and sometimes to say why the model fails for a particular value.
The model breaks at n = 0 because you cannot divide by zero, and in real life nobody does the job with zero workers. Also the number of people has to be a whole number, so an answer such as 2.5 people is not sensible. Remember that the two branches of a reciprocal graph are separate, so you must never join the points on one side to the points on the other.
Method: 1. Find k by multiplying a matching pair of values together. 2. Substitute into y = k/x to find the value asked for. 3. To find x from y, divide k by y. 4. Say whether the answer is sensible, and explain why x = 0 is not allowed.
Tip: Multiply the x and y of any point on a reciprocal curve and you always get k.
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