Graphs of quadratic functions

Every quadratic graph is a parabola with one turning point on its line of symmetry. These six show y = x², its reflection, its translations and the factorised and completed-square forms you meet in exams.

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 = x²

The basic parabola: a U-shaped curve with its vertex (minimum point) at the origin, symmetric about the y-axis.

Open y = x² 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
Line of symmetry	x = 0
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.

Graphs made from this one: y = −x², y = x² − 4, y = (x − 2)², y = x² − 2x − 3, y = 2x² − 4x + 1. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = −x²

An upside-down parabola (∩ shape) with its maximum point at the origin.

Open y = −x² 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	maximum (0, 0)
Asymptotes	none
Line of symmetry	x = 0
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.

y = −x² is y = x² reflected in the x-axis: see the graph of y = x². Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = x² − 4

The parabola y = x² moved down 4 units. It crosses the x-axis at x = ±2, because x² − 4 = (x − 2)(x + 2).

Open y = x² − 4 in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ −4
x-intercepts	(−2, 0) and (2, 0)
y-intercept	(0, −4)
Turning point	minimum (0, −4)
Asymptotes	none
Line of symmetry	x = 0
Symmetry	even function: f(−x) = f(x), symmetric about the y-axis

How to sketch it

1. Mark the x-intercepts (−2, 0) and (2, 0), the y-intercept (0, −4), the turning point (0, −4).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: y ≥ −4. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and reflect it in the y-axis.

y = x² − 4 is y = x² translated down 4 units: see the graph of y = x². Graphs made from this one: y = |x² − 4|. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = (x − 2)²

The parabola y = x² moved 2 units to the right, so its vertex is at (2, 0).

Open y = (x − 2)² in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ 0
x-intercept	(2, 0), where the curve touches the x-axis
y-intercept	(0, 4)
Turning point	minimum (2, 0)
Asymptotes	none
Line of symmetry	x = 2
Symmetry	neither even nor odd

How to sketch it

1. Mark the x-intercept (2, 0), where the curve touches the x-axis, the y-intercept (0, 4), the turning point (2, 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.

y = (x − 2)² is y = x² translated 2 units to the right (replace x with x − 2): see the graph of y = x². Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = x² − 2x − 3

A parabola that factorises as (x + 1)(x − 3). Completing the square gives (x − 1)² − 4, so the vertex is (1, −4).

Open y = x² − 2x − 3 in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ −4
x-intercepts	(−1, 0) and (3, 0)
y-intercept	(0, −3)
Turning point	minimum (1, −4)
Asymptotes	none
Line of symmetry	x = 1
Symmetry	neither even nor odd

How to sketch it

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

y = x² − 2x − 3 is y = x² translated 1 right and 4 down: (x − 1)² − 4: see the graph of y = x². Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = 2x² − 4x + 1

A narrower parabola (a = 2). Completing the square gives 2(x − 1)² − 1, and the quadratic formula gives roots x = 1 ± √2/2.

Open y = 2x² − 4x + 1 in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	y ≥ −1
x-intercepts	(1 − √2/2, 0) ≈ (0.293, 0) and (1 + √2/2, 0) ≈ (1.71, 0)
y-intercept	(0, 1)
Turning point	minimum (1, −1)
Asymptotes	none
Line of symmetry	x = 1
Symmetry	neither even nor odd

How to sketch it

1. Mark the x-intercepts (1 − √2/2, 0) ≈ (0.293, 0) and (1 + √2/2, 0) ≈ (1.71, 0), the y-intercept (0, 1), the turning point (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.

y = 2x² − 4x + 1 is y = x² stretched vertically by scale factor 2, then translated 1 right and 1 down: 2(x − 1)² − 1: see the graph of y = x². 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.