Parabolas in Vertex Form

Learn to write, graph, and analyze quadratic functions in vertex form y = a(x - h)² + k.

Advanced25 min16 problems
Practice Problems

Definition

The vertex form of a quadratic function is:
y=a(x−h)2+ky = a(x - h)^2 + k
where:
  • (h,k)(h, k) is the vertex of the parabola
  • aa determines the width and direction of opening
  • The axis of symmetry is the vertical line x=hx = h
Key properties:
  • 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 than y=x2y = x^2
  • If ∣a∣<1|a| < 1: parabola is wider than y=x2y = x^2

Try it now

What is the vertex of y=(x−4)2+3y = (x - 4)^2 + 3?

Worked Examples

For y=2(x−3)2−5y = 2(x - 3)^2 - 5, find the vertex, axis of symmetry, and direction of opening.
1

Identify hh from (x−h)(x - h)

(x−3)(x - 3) means h=3h = 3
2

Identify kk as the constant

k=−5k = -5
3

Write the vertex

Vertex = (h,k)(h, k) → (3,−5)(3, -5)
4

Find axis of symmetry

x=hx = h → x=3x = 3
5

Determine direction from aa

a=2>0a = 2 > 0 → Opens upward

Common Mistakes

Reading (x+2)2(x + 2)^2 as h=2h = 2

Why it's wrong: Vertex form has a minus sign: (x+2)2=(x−(−2))2(x + 2)^2 = (x - (-2))^2, so h=−2h = -2, not 22.

Correct: Always rewrite to match (x−h)(x - h): if you see (x+2)(x + 2), then h=−2h = -2.

Forgetting that aa affects width, not just direction

Why it's wrong: Students focus only on whether aa is positive or negative.

Correct: ∣a∣>1|a| > 1 makes the parabola narrower; ∣a∣<1|a| < 1 makes it wider.

Errors in completing the square

Why it's wrong: Forgetting to subtract the same value that was added inside the parentheses.

Correct: When adding 99 to complete the square, you must also subtract 99 to keep the equation balanced.

Watch a video explanation

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

Parabolas Explained | Vertex, Focus, Directrix, and Equations

by Understand The Math

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Interactive Visual

Function graph

y = (x - 2)^2 - 3

-8-6-4-202468-505xyV(2, -3)Axis of symmetry

Tap or hover over the graph to read off values.

y = a(x - h)² + ky = x²Axis of symmetry

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Interactive Grapher

-10-50510-10-50510xy

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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−4)2+3y = (x - 4)^2 + 3?

Why It Matters

Vertex form makes it easy to understand quadratic functions:
  • Physics: Projectile motion uses parabolas. The vertex tells you the maximum height and when it occurs.
  • Engineering: Satellite dishes and car headlights use parabolic reflectors. The bulb or receiver sits at the focus, a point on the axis of symmetry inside the curve.
  • Business: Profit functions are often quadratic. The vertex shows maximum profit.
  • Architecture: Parabolic arches distribute weight efficiently. The vertex is the highest point.
Understanding vertex form lets you quickly identify the most important features of any parabola!

Real World Applications

Projectile Motion

When an object is thrown or launched, its height over time follows a parabolic path. Vertex form immediately tells us the maximum height.

Example:

A ball's height is h(t)=−5(t−2)2+20h(t) = -5(t - 2)^2 + 20 meters. The vertex (2,20)(2, 20) tells us the ball reaches its maximum height of 20 meters at t=2t = 2 seconds.

1Try It Yourself

A water fountain shoots water following the path y=−0.5(x−4)2+8y = -0.5(x - 4)^2 + 8, where yy is height in meters and xx is horizontal distance.

What is the maximum height of the water and at what horizontal distance does it occur?

Step 1: Write the mathematical expression

Read the vertex from (h,k)(h, k):

Satellite Dishes

Parabolic dishes reflect incoming signals to a single point, the focus, which lies on the axis of symmetry above the vertex. Engineers use vertex form to design the precise shape.

Example:

A dish with equation y=0.25x2y = 0.25x^2 has its vertex at the origin and its focus 1 unit above it, at (0,1)(0, 1).

2Try It Yourself

An engineer designs a parabolic reflector with equation y=0.1(x−0)2y = 0.1(x - 0)^2. The reflector is 2 meters wide.

How deep is the dish at its edges (at x=±1x = \pm 1 meter)?

Step 1: Write the mathematical expression

Calculate yy when x=1x = 1:

Key Takeaways

  • 1Vertex form is y=a(x−h)2+ky = a(x - h)^2 + k where (h,k)(h, k) is the vertex
  • 2The axis of symmetry is always x=hx = h
  • 3a>0a > 0: opens upward (minimum at vertex); a<0a < 0: opens downward (maximum at vertex)
  • 4∣a∣>1|a| > 1: narrower parabola; ∣a∣<1|a| < 1: wider parabola
  • 5To convert from standard form, complete the square

Frequently Asked Questions

Look at aa: if a>0a > 0, the parabola opens upward so the vertex is a minimum. If a<0a < 0, it opens downward so the vertex is a maximum.
Look at aa: if a>0a > 0, the parabola opens upward so the vertex is a minimum. If a<0a < 0, it opens downward so the vertex is a maximum.
The form (x−h)(x - h) represents a horizontal shift. When h=3h = 3, we write (x−3)(x - 3), which shifts the parabola 3 units right. When h=−2h = -2, we write (x−(−2))=(x+2)(x - (-2)) = (x + 2), shifting 2 units left.
Expand the squared term: a(x−h)2+k=a(x2−2hx+h2)+k=ax2−2ahx+ah2+ka(x - h)^2 + k = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k.

Glossary

Vertex
The turning point of a parabola (its highest or lowest point); the point (h,k)(h, k) in vertex form
Axis of symmetry
The vertical line x=hx = h that divides the parabola into two mirror images
Vertex form
The equation y=a(x−h)2+ky = a(x - h)^2 + k where (h,k)(h, k) is the vertex
Completing the square
A technique to convert standard form to vertex form by creating a perfect square trinomial

Formula Card

Vertex Form

y=a(x−h)2+ky = a(x - h)^2 + k

$(h, k)$ is the vertex; $a$ sets width and direction

Vertex

(h,k)(h, k)

Coordinates of the highest or lowest point

Axis of Symmetry

x=hx = h

Vertical line through the vertex

Complete the Square

x2+bx+(b2)2=(x+b2)2x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2

Convert to vertex form

Today's challengeOne problem, three tries

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