Quadratic, cubic and reciprocal graphs
Quadratic graphs
A has an x² term and no higher power. It is a smooth curve with a line of symmetry:
- a U shape if the x² term is positive, such as y = x² (curve A)
- an ∩ shape if the x² term is negative, such as y = 4 − x² (curve B).
Example: which shape is y = 3 + 2x − x²?
- The x² term is −x², which is negative.
- An ∩ shape
Key terms
Cubic graphs
A has an x³ term and no higher power. y = x³ rises from bottom left to top right, flattening as it passes through the origin.
If the x³ term is negative, such as y = −x³, the shape is reflected: it falls from top left to bottom right. Other cubics, such as y = x³ − 4x, have the same overall direction but with a hump and a dip in the middle.
Example: which of y = x² − 1, y = 2x − 1, y = x³ + 1 and y = 1/x is a cubic?
- Look for the highest power of x.
- y = x³ + 1, because its highest power is x³
Key terms
Reciprocal graphs
A has the form y = k/x, such as y = 1/x. It has two separate curves in opposite corners.
- You can't divide by 0, so there is no point at x = 0.
- The curves get closer and closer to the axes but never touch them. Lines like these are called .
For y = 1/x, the curves are in the top right and bottom left. For a negative k, such as y = −1/x, they are in the top left and bottom right.
Example: work out y on y = 1/x when x = 4, x = 0.5 and x = −2.
- 1 ÷ 4 = 0.25
- 1 ÷ 0.5 = 2
- 1 ÷ −2 = −0.5
Key terms
Exponential graphs (Higher tier only)
An has the form y = kˣ, where x is a power, such as y = 2ˣ.
- It always passes through (0, 1), because k⁰ = 1.
- If k is more than 1, it rises more and more steeply. If k is between 0 and 1, it falls instead.
- It never touches the x-axis: y is always positive.
Exponential graphs model growth and decay, such as compound interest.
Example: complete the table for y = 2ˣ.
| x | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y | ¼ | ½ | 1 | 2 | 4 | 8 |
- 2⁻² = 1/2² = ¼
- 2⁰ = 1
- 2³ = 8
Key terms
Plotting a quadratic from a table
- Work out y for each x in the table, taking care with negatives.
- Plot the points.
- Join them with a smooth curve, not straight lines.
Example: draw y = x² − 2x − 3 for −2 ≤ x ≤ 4.
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y | 5 | 0 | −3 | −4 | −3 | 0 | 5 |
- x = −2: (−2)² − 2 × (−2) − 3 = 4 + 4 − 3 = 5
- x = 1: 1 − 2 − 3 = −4
- Plot and join with a smooth curve
Common mistake
Square a negative x before you subtract: (−2)² is +4. The y-values of a quadratic are symmetrical, so a value that breaks the pattern is probably wrong.