Sheet № 148 · Higher only · AQA · Edexcel · OCR
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 c
§Key definitions
Question:
Sketch the graph of y = x³.
Answer:
S-shaped curve passing through the origin, rising steeply to the right and falling steeply to the left.
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.
§Formulas to memorise
Cubic: y = ax³ + bx² + cx + d — S-shaped curve through the origin if simplified to y = x³
Reciprocal: y = a/x — two branches, asymptotes at x = 0 and y = 0
Exponential: y = aˣ — passes through (0, 1), asymptote at y = 0
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
Sketch the graph of y = x³.
Working:
⚠ 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.