Vertex Form

Q: Why does the sign flip on $h$ but not on $k$?

The vertex sits where the parentheses equal zero. For $(x - 3)^{2}$ that happens at $x = 3$, so a minus on the page means a positive $h$. The $k$ is added after the squaring, straight to the output, so it keeps its sign.

Q: What if there are no parentheses, like $y = x^{2} + 5$?

That is still vertex form with $h = 0$: think of it as $y = (x - 0)^{2} + 5$. The vertex is $(0, 5)$ and the axis of symmetry is the $y$-axis, $x = 0$.

Q: What does the $a$ tell me?

Direction and width. Positive $a$ opens up, negative opens down. If $|a| > 1$ the parabola is narrower than $y = x^{2}$; if $|a| < 1$ it is wider. The $a$ has no effect on where the vertex is.

Vertex form is the quadratic written as $y = a(x - h)^{2} + k$ . Its whole point is speed: the vertex is $(h, k)$ , the axis of symmetry is $x = h$ , and the sign of $a$ tells you which way the parabola opens — all readable in seconds, with no computation.

The one thing that trips students up is the inside sign. The formula says $x - h$ , so the $h$ you read off has the opposite sign of what appears on the page: $(x - 3)^{2}$ means $h = 3$ , and $(x + 3)^{2}$ means $h = -3$ . The outside number $k$ keeps its sign.

Reading the vertex

Match the equation to the template $y = a(x - h)^{2} + k$ . The $h$ is whatever value makes the parentheses zero — flip the sign you see inside. The $k$ is the number added at the end, sign unchanged.

So for $y = (x - 3)^{2} + 5$ : the inside gives $h = 3$ , the outside gives $k = 5$ , and the vertex is $(3, 5)$ . For $y = (x + 2)^{2} - 5$ : the inside gives $h = -2$ , the outside gives $k = -5$ , and the vertex is $(-2, -5)$ .

Axis of symmetry and direction

The axis of symmetry is the vertical line through the vertex: $x = h$ . Fold the parabola along that line and the two halves match exactly.

The $a$ out front works the same as always: $a > 0$ opens up, $a < 0$ opens down, and $|a|$ controls the width — bigger than $1$ is narrower, between $0$ and $1$ is wider.

The graph below is $y = 2(x - 3)^{2} - 4$ : vertex at $(3, -4)$ , axis of symmetry $x = 3$ , opening up and narrower than the parent because $a = 2$ .

Writing an equation in vertex form

To build the equation from a vertex and an $a$ -value, drop $h$ and $k$ into the template. Vertex $(4, -1)$ with $a = 2$ becomes $y = 2(x - 4)^{2} - 1$ . Watch both signs: the $4$ flips to $x - 4$ inside, and the $-1$ stays $-1$ outside.

Worked examples

Example 1: read everything off the equation

Find the vertex, axis of symmetry, and direction of $y = 2(x - 3)^{2} - 4$ .

Match to the template

y = a(x - h)^{2} + k

Inside gives

h

— flip the sign

h = 3

Outside gives

k

— keep the sign

k = -4

Identify

a

— positive, so it opens up

a = 2

Answer: Vertex $(3, -4)$ , axis of symmetry $x = 3$ , opens up.

Example 2: the sign trap

Find the vertex of $y = (x + 2)^{2} - 5$ .

Rewrite the inside as a subtraction

x + 2 = x - (-2)

h

is negative

h = -2

The outside number keeps its sign

k = -5

Answer: Vertex $(-2, -5)$ .

Example 3: write the equation

Write the vertex form equation of a parabola with vertex $(4, -1)$ and $a = 2$ .

Start with the template

y = a(x - h)^{2} + k

Substitute

h = 4

— it flips to minus inside

y = 2(x - 4)^{2} + k

Substitute

k = -1

y = 2(x - 4)^{2} - 1

Answer: $y = 2(x - 4)^{2} - 1$

Try one yourself

Common questions

Why does the sign flip on

h

but not on

k

The vertex sits where the parentheses equal zero. For $(x - 3)^{2}$ that happens at $x = 3$ , so a minus on the page means a positive $h$ . The $k$ is added after the squaring, straight to the output, so it keeps its sign.

What if there are no parentheses, like

y = x^{2} + 5

That is still vertex form with $h = 0$ : think of it as $y = (x - 0)^{2} + 5$ . The vertex is $(0, 5)$ and the axis of symmetry is the $y$ -axis, $x = 0$ .

What does the

a

tell me?

Direction and width. Positive $a$ opens up, negative opens down. If $|a| > 1$ the parabola is narrower than $y = x^{2}$ ; if $|a| < 1$ it is wider. The $a$ has no effect on where the vertex is.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Vertex Form

Reading the vertex

Axis of symmetry and direction

Writing an equation in vertex form

Worked examples

Example 1: read everything off the equation

Example 2: the sign trap

Example 3: write the equation

Try one yourself

Common questions

Keep learning

Translations of Quadratic Functions

Dilations of Quadratic Functions

Introduction to Quadratic Functions

Solve by Graphing