Vertex Form

Learn to write and interpret quadratic functions in vertex form to easily identify the vertex and graph parabolas.

Intermediate25 minLesson

Definition

Vertex form is a way of writing a quadratic function that makes it easy to identify the vertex of the parabola:

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

Where:

$(h, k)$ is the vertex (highest or lowest point)
$a$ determines the direction and width of the parabola
If $a > 0$ : parabola opens upward (vertex is minimum)
If $a < 0$ : parabola opens downward (vertex is maximum)
If $|a| > 1$ : parabola is narrower
If $|a| < 1$ : parabola is wider
The axis of symmetry is the vertical line $x = h$

Try it now

What is the vertex of

y = (x - 2)^2 + 3

Worked Examples

Find the vertex of $y = 2(x - 3)^2 + 5$

Identify the form

This is in vertex form: $y = a(x - h)^2 + k$ → $a = 2$ , $h = 3$ , $k = 5$

Find h (x-coordinate)

In $(x - h)^2$ , we have $(x - 3)^2$ , so $h = 3$

Find k (y-coordinate)

The constant at the end is $k = 5$

Write the vertex

Vertex = $(h, k)$ → $(3, 5)$

Answer: The vertex is $(3, 5)$ . Since $a = 2 > 0$ , the parabola opens upward, so $(3, 5)$ is the minimum point.

Common Mistakes

Getting the sign of h wrong: thinking $(x + 4)^2$ means $h = 4$

Why it's wrong: The formula is $(x - h)^2$ . When you see $(x + 4)^2$ , it's actually $(x - (-4))^2$ .

Correct: If you see $+$ inside the parentheses, $h$ is negative. $(x + 4)^2$ means $h = -4$ .

Confusing which direction the parabola opens

Why it's wrong: Students sometimes think $a$ affects horizontal direction instead of vertical.

Correct: $a > 0$ means opens UP (like a smile). $a < 0$ means opens DOWN (like a frown).

Forgetting that the vertex is a minimum when $a > 0$ and maximum when $a < 0$

Why it's wrong: The vertex is always an extreme point, but which type depends on the direction.

Correct: Opening up = valley = minimum. Opening down = hill = maximum.

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y = 2x + 1

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is the vertex of $y = (x - 2)^2 + 3$ ?

Why It Matters

Vertex form is incredibly useful because it immediately reveals key information about a parabola:

Physics: When you throw a ball, vertex form tells you the maximum height and when it occurs
Business: Finding the price that maximizes profit or minimizes cost
Engineering: Designing parabolic mirrors and satellite dishes that focus at a specific point
Architecture: Creating arches and bridges with precise highest/lowest points

Instead of calculating the vertex from standard form, vertex form gives you the answer directly!

Real World Applications

Projectile Motion

When an object is thrown, its height over time follows a parabola. Vertex form reveals the maximum height.

Example:

A ball's height is $h(t) = -5(t - 2)^2 + 25$ meters. The vertex $(2, 25)$ tells us the ball reaches 25 meters at $t = 2$ seconds.

1Try It Yourself

A rocket's height is given by $h(t) = -4(t - 5)^2 + 100$ meters.

What is the maximum height and when does it occur?

Step 1: Write the mathematical expression

Identify the vertex $(h, k)$ :

Business Optimization

Companies use quadratic functions to model profit. The vertex shows the price that maximizes profit.

Example:

Profit function $P(x) = -2(x - 50)^2 + 5000$ has vertex $(50, 5000)$ . Selling at 50 dollars gives maximum profit of 5000 dollars.

2Try It Yourself

A company's profit is $P(x) = -3(x - 40)^2 + 4800$ dollars, where $x$ is the price.

What price maximizes profit, and what is that profit?

Step 1: Write the mathematical expression

Find the vertex:

Architecture - Parabolic Arches

Architects design arches using parabolas. The vertex determines the highest point of the arch.

Example:

An arch modeled by $y = -0.1(x - 10)^2 + 10$ has its peak at $(10, 10)$ - the center is 10 meters high.

3Try It Yourself

A bridge arch follows $y = -0.05(x - 20)^2 + 15$ meters.

How high is the arch at its peak, and where is the peak located?

Step 1: Write the mathematical expression

Identify the vertex:

Key Takeaways

1Vertex form is $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex
2If $a > 0$ , the parabola opens upward (vertex is minimum)
3If $a < 0$ , the parabola opens downward (vertex is maximum)
4The axis of symmetry is the vertical line $x = h$
5Watch the signs: $(x + 4)^2$ means $h = -4$ , not $h = 4$

Frequently Asked Questions

Use completing the square. For

y = ax^2 + bx + c

, factor out

a

from the first two terms, complete the square, then simplify. Alternatively, find the vertex using

h = -\frac{b}{2a}

and

k = f(h)

Glossary

Vertex: The highest or lowest point on a parabola; the turning point
Axis of symmetry: The vertical line that passes through the vertex, dividing the parabola into two mirror images
Parabola: The U-shaped curve that is the graph of a quadratic function
Vertex form: The form $y = a(x - h)^2 + k$ where the vertex is easily identified as $(h, k)$

Formula Card

Vertex Form

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

Standard vertex form equation

Vertex

(h, k)

Coordinates of the vertex

Axis of Symmetry

x = h

Vertical line through the vertex

Direction

a > 0

: up,

a < 0

: down

Parabola opening direction

Vertex Form

Definition

Worked Examples

Common Mistakes

Interactive Visual

Linear Function Explorer

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Projectile Motion

Business Optimization

Architecture - Parabolic Arches

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Vertex Form

Vertex

Axis of Symmetry

Direction

More in This Topic

Solving Quadratic Equations by Square Roots

Introduction to Quadratic Equations

Completing the Square

The Quadratic Formula