Graphs of cubic and quartic functions

Cubics have up to two turning points and always one point of inflection; quartics have up to three turning points. Factorise to find the roots, differentiate to find the turning points.

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 = x³

The basic cubic: it rises from bottom left to top right and flattens out at the origin, a stationary point of inflection (not a turning point).

Open y = x³ in the Grapher

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), a stationary point of inflection
Asymptotes	none
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Mark the x-intercept (0, 0), the y-intercept (0, 0).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

Graphs made from this one: y = −x³, y = ∛x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = −x³

The cubic y = x³ reflected in the x-axis: it falls from top left to bottom right, flattening at the origin.

Open y = −x³ in the Grapher

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), a stationary point of inflection
Asymptotes	none
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Mark the x-intercept (0, 0), the y-intercept (0, 0).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

y = −x³ is y = x³ reflected in the x-axis (or in the y-axis: for an odd function it is the same): see the graph of y = x³. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = x³ − x

A cubic that factorises as x(x − 1)(x + 1), so it crosses the x-axis three times. Setting dy/dx = 3x² − 1 = 0 gives the turning points at x = ±√3/3.

Open y = x³ − x in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	all real numbers (y ∈ ℝ)
x-intercepts	(−1, 0), (0, 0) and (1, 0)
y-intercept	(0, 0)
Turning points	maximum (−√3/3, 2√3/9) ≈ (−0.577, 0.385); minimum (√3/3, −2√3/9) ≈ (0.577, −0.385)
Point of inflection	(0, 0)
Asymptotes	none
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Mark the x-intercepts (−1, 0), (0, 0) and (1, 0), the y-intercept (0, 0), the turning points (−√3/3, 2√3/9) ≈ (−0.577, 0.385) and (√3/3, −2√3/9) ≈ (0.577, −0.385).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = x³ − 3x

A cubic with turning points at x = ±1 (dy/dx = 3x² − 3 = 0) and roots at 0 and ±√3, since x³ − 3x = x(x² − 3).

Open y = x³ − 3x in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	all real numbers (y ∈ ℝ)
x-intercepts	(−√3, 0), (0, 0) and (√3, 0); √3 ≈ 1.73
y-intercept	(0, 0)
Turning points	maximum (−1, 2); minimum (1, −2)
Point of inflection	(0, 0)
Asymptotes	none
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Mark the x-intercepts (−√3, 0), (0, 0) and (√3, 0); √3 ≈ 1.73, the y-intercept (0, 0), the turning points (−1, 2) and (1, −2).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = x³ − 3x² + 4

A cubic with a repeated root: x³ − 3x² + 4 = (x + 1)(x − 2)², so it crosses the x-axis at −1 and touches it at 2.

Open y = x³ − 3x² + 4 in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	all real numbers (y ∈ ℝ)
x-intercepts	(−1, 0), and (2, 0) where the curve touches the x-axis
y-intercept	(0, 4)
Turning points	maximum (0, 4); minimum (2, 0)
Point of inflection	(1, 2)
Asymptotes	none
Symmetry	neither even nor odd

How to sketch it

1. Mark the x-intercepts (−1, 0), and (2, 0) where the curve touches the x-axis, the y-intercept (0, 4), the turning points (0, 4) and (2, 0).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: all real numbers (y ∈ ℝ). 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 = x⁴

Like y = x² but flatter near the origin and steeper further out. Its minimum is at the origin.

Open y = x⁴ in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ 0
x-intercept	(0, 0), where the curve touches the x-axis
y-intercept	(0, 0)
Turning point	minimum (0, 0)
Asymptotes	none
Symmetry	even function: f(−x) = f(x), symmetric about the y-axis

How to sketch it

1. Mark the x-intercept (0, 0), where the curve touches the x-axis, the y-intercept (0, 0), the turning point (0, 0).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: y ≥ 0. Your sketch should not go anywhere else.
4. 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 = x⁴ − 2x²

A W-shaped quartic: x⁴ − 2x² = x²(x² − 2), with dy/dx = 4x³ − 4x = 0 at x = 0 and x = ±1.

Open y = x⁴ − 2x² in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ −1
x-intercepts	(−√2, 0), (√2, 0), and (0, 0) where the curve touches the x-axis; √2 ≈ 1.41
y-intercept	(0, 0)
Turning points	minimum (−1, −1); maximum (0, 0); minimum (1, −1)
Asymptotes	none
Symmetry	even function: f(−x) = f(x), symmetric about the y-axis

How to sketch it

1. Mark the x-intercepts (−√2, 0), (√2, 0), and (0, 0) where the curve touches the x-axis; √2 ≈ 1.41, the y-intercept (0, 0), the turning points (−1, −1) and (0, 0) and (1, −1).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: y ≥ −1. Your sketch should not go anywhere else.
4. 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.

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.