Graphs of square root and cube root functions
A square root graph stops at the edge of its domain, because you cannot take the square root of a negative number; the cube root is defined for every x. These three show what that does to the domain and range.
Every value in the key-features tables is checked by script against the Grapher's own calculations (roots, turning points, asymptotes, domain and range).
Graph of y = √x
Half of a sideways parabola: it starts at the origin and keeps rising, ever more slowly. It is the inverse of y = x² for x ≥ 0.
Key features
| Domain | x ≥ 0 |
|---|---|
| Range | y ≥ 0 |
| x-intercept | (0, 0) |
| y-intercept | (0, 0) |
| Turning points | none |
| Asymptotes | none |
| Symmetry | neither even nor odd |
The curve is vertical where it starts: the gradient 1/(2√x) grows without limit as x → 0.
How to sketch it
- Mark the x-intercept (0, 0), the y-intercept (0, 0).
- Only draw the curve where it is defined: x ≥ 0.
- Check the range: y ≥ 0. Your sketch should not go anywhere else.
Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = √(4 − x²)
The upper half of the circle x² + y² = 4 (centre the origin, radius 2).
Key features
| Domain | −2 ≤ x ≤ 2 |
|---|---|
| Range | 0 ≤ y ≤ 2 |
| x-intercepts | (−2, 0) and (2, 0) |
| y-intercept | (0, 2) |
| Turning point | maximum (0, 2) |
| Asymptotes | none |
| Symmetry | even function: f(−x) = f(x), symmetric about the y-axis |
How to sketch it
- Mark the x-intercepts (−2, 0) and (2, 0), the y-intercept (0, 2), the turning point (0, 2).
- Only draw the curve where it is defined: −2 ≤ x ≤ 2.
- Check the range: 0 ≤ y ≤ 2. Your sketch should not go anywhere else.
- Use the symmetry: draw the right-hand half and reflect it in the y-axis.
Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = ∛x
The inverse of y = x³: defined for every x (the cube root of a negative number is negative), with a vertical tangent at the origin.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | all real numbers (y ∈ ℝ) |
| x-intercept | (0, 0) |
| y-intercept | (0, 0) |
| Turning points | none |
| Point of inflection | (0, 0), where the tangent is vertical |
| Asymptotes | none |
| Symmetry | odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin |
How to sketch it
- Mark the x-intercept (0, 0), the y-intercept (0, 0).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
- Use the symmetry: draw the right-hand half and rotate it 180° about the origin.
y = ∛x is the reflection of y = x³ in the line y = x (its inverse): see the graph of y = x³. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Notes and more graphs
Other families of graphs
- Graphs of straight lines
- Graphs of quadratic functions
- Graphs of cubic and quartic functions
- Graphs of reciprocal and rational functions
- Graphs of modulus (absolute value) functions
- Graphs of exponential and logistic functions
- Graphs of logarithmic functions
- Graphs of trigonometric functions
- Graphs of inverse trigonometric functions
All graphs of common functions · Graph transformations · Open the Grapher
The Grapher is our own free tool (graph.mathrevision.co.uk), built for IB, A Level and IGCSE students. It is not an exam calculator: practise on your own GDC for calculator papers.