# Formulas, graphs & relations » Vertices of parabolas

## Contents

Preface: how do you calculate`y`

_{vertex}?

Symmetry

`x`

_{vertex}= –

`b`(2

`a`)

How is

`x`

_{vertex}= –

`b`(2

`a`) derived?

Reasoning, finding (

`p`,

`q`) using the formula

## Preface: how do you calculate `y`_{vertex}?

In the theory below you will find an extensive explanation about finding the `x`_{vertex}.

To calculate `y`_{vertex} you first need to calculate `x`_{vertex}.

Once you know `x`_{vertex}, fill in the value of `x`_{vertex} into the formula to calculate `y`_{vertex}.

In function notation: `y`_{vertex} = `f` (`x`_{vertex}).

## Symmetry

The quadratic function `f` (`x`) = `a`(`x` – `α`)(`x` – `β`) intersects the `x`-axis always in `α` and `β`.

So `x`_{vertex} = `α` + `β`2 = lies precisely in between `α` and `β`.

In other words:

If you know two points that are exactly on the same height, then `x`_{vertex} lies precisely in between.

*Example 1*

It is known that a parabola has points (4, 13) and (12, 13). What is `x`_{vertex}?

Answer:

The axis of symmetry lies exactly between `x` = 4 and `x` = 12.

The `x`_{vertex} is on this axis of symmetry.

So `x`_{vertex} = 4 + 122 = 8.

*Example 2*

Calculate the coordinates of the vertex of the parabola that

corresponds to `f` (`x`) = 2(`x` + 3)(`x` – 7).

Answer:

From the function you can derive that the parabola will intersect the `x`-axis at `x` = –3 and `x` = 7. Solve the equation 2(`x` + 3)(`x` – 7) = 0, if you do not see this.

So `x`_{vertex} = –3 + 72 = 2

Then `y`_{vertex} = 2(2 + 3)(2 – 7) = –50.

Coordinates are (2, –50).

`x`_{vertex} = –`b`(2`a`)

The `x`-coordinate of the vertex of the graph that corresponds to function `f` (`x`) = `ax`^{2} + `bx` + `c` with `a` ≠ 0 can be calculated using this formula:

`x`_{vertex} = –`b`2`a`

*Example 1*

Given is `y` = –4`x`^{2} + 8`x` – 5.

Calculate the coordinates of the vertex of the corresponding graph.

Answer:

`x`_{vertex} = –82 × –4 = 1

Then `y`_{vertex} = –4 × 1^{2} + 8 × 1 – 5 = –1.

Coordinates are (1, –1).

*Example 2*

Given is function `f` (`x`) = 2`x`^{2} + `px` + 3 of which the minimum is 1.

Calculate `p` algebraically.

Answer:

First make the formula for `x`_{vertex}:

`x`_{vertex} = –`p`2 × –2 = –14`p`

If we substitute –14`p` for `x` in the formula than the outcome must be the minimum 1.

That gives us the following equation which can be solved:

2 × (–14p)^{2} + p × –14p + 3 | = 1^{ } |

2 × 116p^{2} + –14p^{2} + 3 | = 1^{ } |

18p^{2} – 14p^{2} + 3 | = 1^{ } |

–18p^{2} + 3 | = 1^{ } |

18p^{2} | = –2^{ } |

p^{2} | = 16^{ } |

p = –4 | of p = 4 |

## How is `x`_{vertex} = –`b`(2`a`) derived?

Every quadratic equation can be written as `ax`^{2} + `bx` + `c` = 0.

However this equation does not always have solutions. The equation `ax`^{2} + `bx` + `c` = `c` always has solutions, as there always is an intersection with the vertical axis. At the first section of this page, you learned you can calculate the vertex using symmetry. So we have to take the average of the solutions of this equation.

ax^{2} + bx + c | = c^{ } | |

ax^{2} + bx | = 0^{ } | |

x(ax + b) | = 0 | |

x = 0 of ax | + b | = 0 |

ax | = –b | |

x | = –ba |

Now we take the average of these to points that lie at the same height.

`x`_{vertex} = 0 + –`b``a`2 = –`b``a`2 = –`b`2`a`

## Reasoning, finding (`p`, `q`) using the formula

The vertex of the graph corresponding to function `f` (`x`) = `a`(`x` – `p`)^{2} + `q` with `a` ≠ 0 is point (`p` , `q`). This is vecause when `x` = `p` you will have (`x` – `p`)^{2} = 0.

*Example 1*

Given is `f` (`x`) = 2(`x` – 3)^{2} – 4.

Give the coordinates of the vertex.

Answer:

`p` = 3 and `q` = –4 so the vertex is (3, –4).

*Example 2*

Given is `f` (`x`) = 7(3`x` + 9)^{2} + 8.

Give the coordinates of the vertex.

Answer:

Calculating `p` costs a little bit more work.
You now have to solve 3`x` + 9 = 0.

3x + 9 | = 0 |

3x | = –9 |

x | = –3 |

`p` = –3 and `q` = 8 so the vertex is (–3, 8).