Cubic and Reciprocal Graphs
At Higher level you need to recognise and sketch graphs beyond straight lines and parabolas. Cubic graphs (y = x³), reciprocal graphs (y = 1/x), and exponential graphs (y = aˣ) each have distinctive shapes. Exam questions often show you a graph and ask you to match it to an equation, or they ask you to sketch a given function. This page covers the key shapes and features you need to know.
What Are Cubic, Reciprocal, and Exponential Graphs?
A cubic graph has an equation where the highest power of x is 3, such as y = x³ or y = 2x³ − 3x. It produces an S-shaped curve.
A reciprocal graph has the form y = a/x (or equivalently y = ax⁻¹). The curve has two branches that approach but never touch the axes — these are called asymptotes.
An exponential graph has the form y = aˣ where a > 0. The curve grows rapidly in one direction and approaches y = 0 in the other.
Key Formulas
Step-by-Step Method
- Identify the type of equation from the highest power or form (x³ for cubic, 1/x for reciprocal, aˣ for exponential).
- Recall the standard shape for that type (see descriptions above).
- Find key features: intercepts, asymptotes, and the general direction of the curve.
- Plot a few key points if needed (e.g., x = −2, −1, 0, 1, 2).
- Draw a smooth curve through the points with the correct shape.
Worked Example 1 — Foundation Level
Question: Sketch the graph of y = x³.
Working:
Key points: (−2, −8), (−1, −1), (0, 0), (1, 1), (2, 8).
The curve passes through the origin, goes from bottom-left to top-right in an S-shape. It is steep for large values of x and flat near the origin.
Answer: S-shaped curve passing through the origin, rising steeply to the right and falling steeply to the left.
Worked Example 2 — Higher Level
Question: Sketch the graph of y = 1/x and state the asymptotes.
Working:
Key points: (−2, −0.5), (−1, −1), (−0.5, −2), (0.5, 2), (1, 1), (2, 0.5).
When x is positive, y is positive (curve in the first quadrant). When x is negative, y is negative (curve in the third quadrant). The curve never touches the x-axis or the y-axis.
Asymptotes: x = 0 (the y-axis) and y = 0 (the x-axis).
Answer: Two branches in opposite quadrants (quadrants 1 and 3), with asymptotes at x = 0 and y = 0.
Worked Example 3 — Exam Style
Question: Match each equation to its graph: (a) y = x³ − 3x, (b) y = 3/x, (c) y = 2ˣ.
Working:
- y = x³ − 3x is a cubic — S-shaped with a local maximum and minimum. It crosses the x-axis at x = 0, x = √3, and x = −√3.
- y = 3/x is a reciprocal — two branches in quadrants 1 and 3, not touching the axes.
- y = 2ˣ is exponential — passes through (0, 1), rises steeply to the right, approaches y = 0 to the left.
Answer: Match by recognising the S-shape (cubic), the two-branch hyperbola (reciprocal), and the rapid growth curve through (0, 1) (exponential).
Common Mistakes
- Confusing cubic and quadratic shapes. A quadratic is U-shaped or ∩-shaped. A cubic is S-shaped. They look very different.
- Drawing the reciprocal curve touching the axes. The curve y = 1/x never touches or crosses either axis. The axes are asymptotes — the curve approaches them but never reaches them.
- Forgetting that y = aˣ always passes through (0, 1). Since a⁰ = 1 for any positive a, the y-intercept is always 1.
Exam Tips
- In graph recognition questions, look for the distinctive features: S-shape = cubic, two separate branches = reciprocal, rapid growth through (0, 1) = exponential.
- If a cubic has a negative leading coefficient (e.g., y = −x³), the S-shape is reflected — it goes from top-left to bottom-right.
- For y = −1/x, the branches move to quadrants 2 and 4 instead of 1 and 3.
- Label asymptotes on reciprocal graphs — this often earns a mark.
Practice Questions
Q1 (Foundation): Plot the points and sketch y = x³ for x = −2 to x = 2.
Q2 (Higher): State the asymptotes of y = 5/x.
Q3 (Higher): Sketch y = 3ˣ and state the y-intercept.
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Related Topics
Summary
- Cubic graphs (y = x³) have an S-shaped curve that passes through the origin.
- Reciprocal graphs (y = 1/x) have two branches with asymptotes at x = 0 and y = 0.
- Exponential graphs (y = aˣ) pass through (0, 1) and have the asymptote y = 0.
- Recognise each graph type by its distinctive shape in exam questions.
- A negative coefficient reflects the standard shape across the x-axis.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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