AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Graphs: including simple cubic functions and the reciprocal function y = 1/x with x ≠ 0
Recognise, sketch and interpret graphs of simple cubic functions and the reciprocal function y = 1/x (where x ≠ 0).
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Recognise, sketch and interpret graphs of simple cubic functions and the reciprocal function y = 1/x (where x ≠ 0). A simple cubic, y = ax³ + bx² + cx + d (a ≠ 0), has either two or no turning points. For example, y = x³ passes through the origin, rising steeply for positive x and falling steeply for negative x, with a stationary point of inflection at the origin, not a turning point. The reciprocal graph y = 1/x has two separate branches. As x approaches zero, y becomes very large, making the y-axis a vertical asymptote. As x becomes very large or small, y approaches zero, making the x-axis a horizontal asymptote. The domain excludes x = 0. You must be able to match these equations to their corresponding graphs.
What earns the marks
- Recognises that a simple cubic function has the form y = ax³ + bx² + cx + d with a ≠ 0 and can sketch a typical cubic shape with two or no turning points.
- Identifies the reciprocal function y = 1/x and states that x ≠ 0, so the graph has no y-intercept and is undefined at x = 0.
- Describes the two branches of y = 1/x in the correct quadrants: for x > 0, y > 0, and for x < 0, y < 0.
- States that the x-axis and y-axis are asymptotes of y = 1/x, and explains that the curve approaches but never touches them.
- Matches a given equation or graph to its correct family, distinguishing a cubic from a reciprocal graph by shape and key features.
Common mistakes and how to put them right
Thinking that y = 1/x can take the value x = 0.
Put it right: The function is undefined there, so the graph never crosses the y-axis and has a vertical asymptote at x = 0.
Drawing y = 1/x as a single continuous curve through the origin.
Put it right: The correct graph has two separate branches in opposite quadrants and does not pass through the origin.
Assuming every cubic graph has exactly two turning points.
Put it right: A real cubic has either two or no local turning points (a stationary point of inflection, as in y = x³, is not a turning point).
Exam tips
- When sketching y = 1/x, draw both branches and label the asymptotes clearly; examiners look for the correct quadrants and the approach to the axes.
- For cubic graphs, mark any intercepts you can find and show the general shape; do not waste time plotting many points if the question only asks for a sketch.
- Check whether the question restricts the domain, especially for y = 1/x where x ≠ 0; state this restriction if asked to give the domain.
Try the lesson
The start of the Graphs: including simple cubic functions and the reciprocal function y = 1/x with x ≠ 0 lesson, made from checked teaching notes for this specification point.
Have a go first
For the reciprocal graph y = 1/x, what is the value of y when x = 4?
Listen
Recognise, sketch and interpret graphs of simple cubic functions and the reciprocal function y = 1/x (where x ≠ 0). A simple cubic, y = ax³ + bx² + cx + d (a ≠ 0), has either two or no turning points. For example, y = x³ passes through the origin, rising steeply for positive x and falling steeply for negative x, with a stationary point of inflection at the origin, not a turning point. The reciprocal graph y = 1/x has two separate branches. As x approaches zero, y becomes very large, making the y-axis a vertical asymptote. As x becomes very large or small, y approaches zero, making the x-axis a horizontal asymptote. The domain excludes x = 0. You must be able to match these equations to their corresponding graphs.
Key words
Say what each word means before you turn the card.
simple cubic function
Try answering aloud or in your head before turning the card.
reciprocal function
Try answering aloud or in your head before turning the card.
asymptote
Try answering aloud or in your head before turning the card.
domain
Try answering aloud or in your head before turning the card.
See every definition
- simple cubic function
- A function of the form y = ax³ + bx² + cx + d with a ≠ 0, which has either two or no turning points.
- reciprocal function
- The function y = 1/x with x ≠ 0, whose graph has two separate branches and does not pass through the origin.
- asymptote
- A line such as the x-axis or y-axis that the curve approaches but never touches.
- domain
- The set of values the function is defined for; for y = 1/x it excludes x = 0.
There is more to this lesson
The full lesson goes on to check your understanding and help you earn the marks. Create a free account to carry on, with your progress saved.
7 days Premium · Then the free plan continues · No card, no charge
Test your understanding of Graphs: including simple cubic functions and the reciprocal function y = 1/x with x ≠ 0
Active recall beats re-reading. Take an instant quiz now to test what you’ve learned and lock it into long-term memory.
7 days Premium · Then the free plan continues · No card, no charge