Quadratic Functions

The Parabola

Graphing a quadratic function and identifying its vertex, axis of symmetry, and direction.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

The Shape Behind x²

Before reading on, plot a few points of y = x² by hand: try x = −2, −1, 0, 1, 2. What do you notice about the y-values for x = −2 and x = 2? For x = −1 and x = 1?

Squaring a number and squaring its opposite always give the same result, since (−a)² = a² — which is exactly why x = −2 and x = 2 both give y = 4, and x = −1 and x = 1 both give y = 1. This built-in symmetry around x = 0 is what gives the graph of any quadratic function its distinctive curved, mirror-image shape.

Definition — Parabola, Vertex, and Axis of Symmetry

The graph of a quadratic function y = ax² + bx + c (with a ≠ 0) is a U-shaped curve called a parabola. Its turning point is the vertex, and the vertical line through the vertex, x = −b/(2a), is the axis of symmetry — the graph is a mirror image of itself across this line.

The leading coefficient, a, controls two visible features: its sign determines whether the parabola opens upward (a positive) or downward (a negative), and its size controls how narrow or wide the curve looks — a larger |a| makes it narrower, since y grows faster for the same x-distance from the vertex.

Worked Example — Finding the Vertex from Standard Form

Find the vertex of y = 2x² − 8x + 3. The axis of symmetry is x = −b/(2a) = −(−8)/(2·2) = 8/4 = 2. Substitute x = 2 back into the equation to find the y-coordinate: y = 2(4) − 8(2) + 3 = 8 − 16 + 3 = −5. The vertex is (2, −5).

Worked Example — Determining Opening Direction and Width

Compare y = 3x² and y = −(1/2)x². The first has a = 3, positive, so it opens upward, and |a| = 3 is relatively large, so it's narrow. The second has a = −1/2, negative, so it opens downward, and |a| = 1/2 is small, so it's wider than a typical parabola.

x = -\dfrac{b}{2a}

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

Once the vertex and axis of symmetry are known, one more point on either side (found by substituting a nearby x-value) is often enough to sketch a reasonably accurate parabola by hand, using the mirror symmetry to reflect it to the other side.

Common Mistakes

Forgetting the negative sign in the axis-of-symmetry formula, computing x = b/(2a) instead of x = −b/(2a).
The formula includes a negative sign in front of b — leaving it out shifts the calculated vertex to the wrong side of the y-axis.
Assuming a larger leading coefficient always makes a parabola open more widely, when it actually makes it narrower.
A larger |a| makes y grow faster moving away from the vertex, which pulls the curve inward, not outward — larger |a| means a narrower parabola, not a wider one.

Key Takeaways

A quadratic function's graph is a parabola, symmetric because squaring a number and its opposite give the same result.
The vertex's x-coordinate is −b/(2a); substituting it back into the equation finds the y-coordinate.
The sign of a determines opening direction; the size of |a| determines width.

Summary

Standard form reveals a parabola's opening direction and width directly, though finding its vertex takes a formula. The next lesson introduces a form that shows the vertex immediately, with no formula required at all.

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