How to Write Quadratic Equations in Vertex Form and Standard Form

Q: When should you use vertex form instead of standard form?

Use vertex form when you already know the vertex, the turning point of the parabola. You can substitute it straight in and then only need one more point to find a. Use standard form when you do not know the vertex but have three points on the curve.

Q: How many points do you need to form a quadratic equation?

It depends on the form. In vertex form, the vertex plus one other point is enough, because a is the only unknown left. In standard form, you need three distinct points, since there are three unknowns to find: a, b and c.

Q: How are quadratic equations used in real life?

A real-life shape such as a tunnel can follow a parabola, so a quadratic equation can describe its height. Once you have the equation, you can work out the height at any width. This lets you check whether something, like a lorry, will fit through.

Q: What is the vertex of a parabola?

The vertex is the highest or lowest point of a parabola, also called the turning point. In the vertex form y = a(x - h)² + k, it is the point (h, k). In the tunnel example, the highest point (0, 5) is the vertex you read into the equation.

Key concept

Vertex form writes a quadratic as y = a(x − h)² + k, where (h, k) is the vertex. Once you know the vertex, substitute one more point to find a. Standard form, y = ax² + bx + c, is used instead when you know three points.

How to Write Quadratic Equations in Vertex Form and Standard Form - introduction visual

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Flashcards

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Quadratic equations with vertex form y = a(x - h)² + k and standard form y = ax² + bx + c, labelled as methods for forming quadratic equations.

Forming a quadratic equation using vertex form with equation y = -2(x - 2)² + 3, where the vertex is (2,3) and the curve passes through (3,1).

Forming quadratic equations using the standard form y = 2x² - x + 3, solving coefficients with three given points: (0,3), (1,4), and (2,9).

Solving the classic tunnel problem. The parabola equation is y = -5/9 · x² + 5 and the vertex is at (0,5). When x = 1, the lorry can pass through.

Forms of Quadratic Equations

Quadratics can be written in vertex form or standard form.
Use vertex form when you know the vertex, and standard form when you know points on the parabola.

Practice Questions

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Q1Easy

The vertex of a parabola is at (1, 2) and $a = 1$ . What is the quadratic equation in vertex form?

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When should you use vertex form instead of standard form?

Use vertex form when you already know the vertex, the turning point of the parabola. You can substitute it straight in and then only need one more point to find a. Use standard form when you do not know the vertex but have three points on the curve.

How many points do you need to form a quadratic equation?

It depends on the form. In vertex form, the vertex plus one other point is enough, because a is the only unknown left. In standard form, you need three distinct points, since there are three unknowns to find: a, b and c.

Why does the point where x = 0 give the value of c?

When you substitute x = 0 into y = ax² + bx + c, the terms ax² and bx both become zero. The equation reduces to y = c, so the value of y at the point where x = 0 is exactly the value of c.

How are quadratic equations used in real life?

A real-life shape such as a tunnel can follow a parabola, so a quadratic equation can describe its height. Once you have the equation, you can work out the height at any width. This lets you check whether something, like a lorry, will fit through.

What is the vertex of a parabola?

The vertex is the highest or lowest point of a parabola, also called the turning point. In the vertex form y = a(x - h)² + k, it is the point (h, k). In the tunnel example, the highest point (0, 5) is the vertex you read into the equation.

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