Quadratic Functions in Vertex Form (A Clear Guide for Everyone)

Parabolas in sight: The Clifton Suspension Bridge, Bristol, United Kingdom.
Parabolas in sight: The Clifton Suspension Bridge, Bristol, United Kingdom.

A quadratic function is a function whose graph is a parabola (a U-shaped curve). One of the most useful ways to write a quadratic is in vertex form, because it shows the parabola’s turning point immediately.

1) The vertex form

A quadratic function in vertex form is written as:

f(x) = a(x - h)2 + k

This form is especially helpful because the values h and k tell you the vertex directly.

2) The vertex (turning point)

The vertex is the point where the parabola changes direction.

In vertex form:

Vertex = (h, k)

  • If the parabola opens up, the vertex is the lowest point (a minimum).
  • If the parabola opens down, the vertex is the highest point (a maximum).

3) What the number a does

The number a controls two key things: the direction the parabola opens, and how wide or narrow it is.

  • a > 0 means the parabola opens up (U-shape).
  • a < 0 means the parabola opens down (upside-down U).
  • |a| > 1 makes the parabola narrower (steeper).
  • 0 < |a| < 1 makes the parabola wider (flatter).

So a controls both the “opening direction” and the steepness.

4) What h does (left and right shift)

The number h moves the parabola left or right.

This part can feel confusing at first because it is inside brackets:

  • (x - 3) shifts the graph right by 3.
  • (x + 3) shifts the graph left by 3.

A reliable rule is: the vertex’s x-coordinate is h, so the axis of symmetry is the vertical line:

x = h

5) What k does (up and down shift)

The number k moves the parabola up or down.

  • +k shifts the graph up by k.
  • -k shifts the graph down by |k|.

Also, notice that when x = h, we get:

f(h) = a(0)2 + k = k

So k is the y-value of the vertex (the minimum or maximum value of the function).

6) A worked example

Example:

f(x) = 2(x - 3)2 - 5

  • a = 2: opens up, and it is narrower than y = x2.
  • h = 3: shifts right by 3, so the vertex has x-coordinate 3.
  • k = -5: shifts down by 5, so the vertex has y-coordinate -5.

So the vertex is:

(3, -5)

And the axis of symmetry is:

x = 3

7) Why vertex form is useful

Vertex form is popular because it makes important features easy to read:

  • You can identify the vertex instantly: (h, k).
  • You can identify the minimum or maximum value instantly: it is k.
  • You can write the axis of symmetry instantly: x = h.
  • You can quickly understand how the graph shifts as h and k change.

8) Quick summary

  • Vertex form: f(x) = a(x - h)2 + k
  • Vertex: (h, k)
  • Axis of symmetry: x = h
  • Opens up/down: depends on the sign of a
  • Narrow/wide: depends on |a|

To practise, try rewriting different quadratics into vertex form and then reading the vertex and axis of symmetry from the result.

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