Vertex Form and How to Graph a Parabola

Every quadratic graphs as a U-shaped curve called a parabola. Its turning point, the vertex, is the most useful point on the graph, and vertex form puts that point right in the equation.

L'idée en bref

In vertex form y = a(x − h)² + k, the vertex is (h, k). From standard form y = ax² + bx + c, the vertex sits at x = −b/(2a).

Les étapes

1. Find the x-coordinate of the vertex with x = −b/(2a), or read h from vertex form.
2. Substitute that x into the equation to get the y-coordinate.
3. Check the sign of a: positive opens up, negative opens down.
4. Plot the vertex, the y-intercept, and any x-intercepts, then draw a smooth symmetric curve.

Exemple résolu

Graph y = x² − 4x + 3 and write it in vertex form.

Solution complète

Explication étape par étape

Étape 1

Here a = 1 and b = −4, so the vertex is at x = 4/2 = 2.

Étape 2

Substitute x = 2: y = 4 − 8 + 3 = −1. The vertex is (2, −1).

Étape 3

So y = (x − 2)² − 1. It opens up, crosses the y-axis at 3, and crosses the x-axis at x = 1 and x = 3.

Erreur fréquente

Watch the sign of h. In y = (x + 3)² − 2 the vertex is at x = −3, not 3, because x + 3 is the same as x − (−3).

Exercices d'entraînement

1. Find the vertex of y = (x + 1)² + 4.
2. Find the vertex of y = x² + 6x + 5.
3. Does y = −2(x − 1)² + 3 open up or down, and what is its maximum value?

Corrigés

1. (−1, 4)
2. (−3, −4)
3. It opens down; the maximum value is 3, at x = 1.

Demandez à PennPaper de vous l'expliquer en direct

Si les étapes vous semblent claires mais que vous bloquez encore, lancez une séance PennPaper et demandez au tuteur de dessiner le problème sur le tableau blanc. Voir les symboles bouger un à un, c'est souvent ce qui fait tilt.

Guides associés

• La formule des racines (discriminant) expliquée
• Completing the Square Explained Step by Step
• Comment tracer la représentation graphique d'une fonction affine