Graphs
Grade 7-8Cubic & Reciprocal Graphs
Cubic and reciprocal graphs each have a distinctive shape that is worth recognising on sight: cubics form an S-shaped curve that can cross the x-axis up to three times, while reciprocal graphs split into two curved branches that never touch the axes. This lesson covers substituting values into both types of equation, finding the roots of a cubic graph, and identifying the asymptotes and shape of a reciprocal graph.
What you need to know
- A cubic has an x cubed term and can have up to two turning points and up to three roots.
- A positive cubic runs from bottom-left to top-right; a negative one is the mirror image.
- The reciprocal graph y = 1/x has two separate branches and never touches the axes.
- For a reciprocal graph the axes are asymptotes: the curve gets close but never reaches them.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Substituting values into a cubic equation
Substitute the given value of x into the equation and work through the arithmetic carefully, taking particular care when cubing a negative number, since an odd power keeps the negative sign.
For y = x³ - 4x, when x = 3: y = 3³ - 4(3) = 27 - 12 = 15.
Substituting values into a reciprocal equation
A reciprocal equation in the form y = k/x is evaluated by dividing the constant k by the given value of x.
For y = 12/x, when x = 4: y = 12 ÷ 4 = 3.
Finding where a cubic graph crosses the x-axis
A cubic graph can cross the x-axis up to three times. Each crossing point is a root of the equation, found where the curve has y = 0. On a graph, read off the x-coordinate at each point the curve touches the horizontal axis.
The graph of y = x³ - 4x crosses the x-axis at x = -2, x = 0 and x = 2, giving the points (-2, 0), (0, 0) and (2, 0).
Recognising reciprocal graphs and their asymptotes
A reciprocal graph y = k/x forms two separate curved branches that get closer and closer to the x-axis and y-axis without ever touching them. These lines, x = 0 and y = 0, are called asymptotes. When k is positive, the branches sit in the top-right and bottom-left; when k is negative, they sit in the top-left and bottom-right.
The graph of y = 8/x has asymptotes x = 0 and y = 0, with branches in the top-right and bottom-left, since 8 is positive.
Worked examples
Three exam-style questions, fully solved
y = 12/x. Work out the value of y when x = 4.
Easy- 1.Substitute x = 4 into the equation: y = 12 ÷ 4
Answer: 3
The graph shows y = x³ - 4x. Write down the coordinates of the three points where the curve crosses the x-axis.
Medium- 1.Find where the curve touches the x-axis on the left side: x = -2
- 2.Find where the curve touches the x-axis at the origin: x = 0
- 3.Find where the curve touches the x-axis on the right side: x = 2
Answer: (-2, 0), (0, 0) and (2, 0)
The graph shows y = 8/x for x ≠ 0. Write down the equations of the two asymptotes of the curve.
Hard- 1.Identify the vertical line the curve never touches: x = 0
- 2.Identify the horizontal line the curve never touches: y = 0
Answer: x = 0 and y = 0
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
What is the greatest number of times a cubic graph can cross the x-axis?
The graph of y = 1/x has a vertical asymptote. Write its equation.
Work out the value of y = 8/x when x = 2.
For the graph y = x³, work out y when x = −2.
For the graph y = x³ − 5x, work out y when x = 3.
How many times does the graph of y = 1/x cross the x-axis?
Avoid these
The mistakes students make most often
Making an arithmetic error when cubing a negative number, for example treating (-2)³ as 8 instead of the correct value, -8.
Confusing a reciprocal equation with a cubic equation, and applying the wrong method or expecting the wrong graph shape.
Missing one of the roots when a cubic crosses the x-axis three times, especially the root at x = 0, which is easy to overlook.
Giving only the axis values, like "0 and 0", instead of writing the full equations of the asymptotes, "x = 0 and y = 0".
Confusing which quadrants a reciprocal graph's branches sit in, forgetting that a negative constant flips the branches into the top-left and bottom-right.
FAQ
Questions students and parents ask
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