Vertex Form

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

Intermediate25 minLesson
Printable worksheet

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+ky = a(x - h)^2 + k
Where:
  • (h,k)(h, k) is the vertex (highest or lowest point)
  • aa determines the direction and width of the parabola
  • If a>0a > 0: parabola opens upward (vertex is minimum)
  • If a<0a < 0: parabola opens downward (vertex is maximum)
  • If ∣a∣>1|a| > 1: parabola is narrower
  • If ∣a∣<1|a| < 1: parabola is wider
  • The axis of symmetry is the vertical line x=hx = h

Try it now

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

Worked Examples

Find the vertex of y=2(x−3)2+5y = 2(x - 3)^2 + 5
1

Identify the form

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

Find h (x-coordinate)

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

Find k (y-coordinate)

The constant at the end is k=5k = 5
4

Write the vertex

Vertex = (h,k)(h, k) → (3,5)(3, 5)

Common Mistakes

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

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

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

Confusing which direction the parabola opens

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

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

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

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Quadratic Functions Explained | Vertex, Axis of Symmetry, and Graphing

by Understand The Math

Watch on YouTube

Interactive Visual

Function graph

-8-6-4-202468-10-5051015xy(3, 5)

Tap or hover over the graph to read off values.

y = a(x - h)² + k
  • Vertex: (3, 5)

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 2(x - 3)^2 + 5

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the vertex of y=(x−2)2+3y = (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+25h(t) = -5(t - 2)^2 + 25 meters. The vertex (2,25)(2, 25) tells us the ball reaches 25 meters at t=2t = 2 seconds.

1Try It Yourself

A rocket's height is given by h(t)=−4(t−5)2+100h(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)(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+5000P(x) = -2(x - 50)^2 + 5000 has vertex (50,5000)(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+4800P(x) = -3(x - 40)^2 + 4800 dollars, where xx 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+10y = -0.1(x - 10)^2 + 10 has its peak at (10,10)(10, 10) - the center is 10 meters high.

3Try It Yourself

A bridge arch follows y=−0.05(x−20)2+15y = -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+ky = a(x - h)^2 + k where (h,k)(h, k) is the vertex
  • 2If a>0a > 0, the parabola opens upward (vertex is minimum)
  • 3If a<0a < 0, the parabola opens downward (vertex is maximum)
  • 4The axis of symmetry is the vertical line x=hx = h
  • 5Watch the signs: (x+4)2(x + 4)^2 means h=−4h = -4, not h=4h = 4

Frequently Asked Questions

Use completing the square. For y=ax2+bx+cy = ax^2 + bx + c, factor out aa from the first two terms, complete the square, then simplify. Alternatively, find the vertex using h=−b2ah = -\frac{b}{2a} and k=f(h)k = f(h).
Use completing the square. For y=ax2+bx+cy = ax^2 + bx + c, factor out aa from the first two terms, complete the square, then simplify. Alternatively, find the vertex using h=−b2ah = -\frac{b}{2a} and k=f(h)k = f(h).
Yes! If ∣a∣<1|a| < 1 (like a=0.5a = 0.5 or a=12a = \frac{1}{2}), the parabola is wider. If ∣a∣>1|a| > 1, it's narrower.
Then a=1a = 1. For example, y=(x−3)2+2y = (x - 3)^2 + 2 has a=1a = 1.

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+ky = a(x - h)^2 + k where the vertex is easily identified as (h,k)(h, k)

Formula Card

Vertex Form

y=a(x−h)2+ky = a(x - h)^2 + k

Standard vertex form equation

Vertex

(h,k)(h, k)

Coordinates of the vertex

Axis of Symmetry

x=hx = h

Vertical line through the vertex

Direction

a>0a > 0: up, a<0a < 0: down

Parabola opening direction

Today's challengeOne problem, three tries

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