Parabolas in Vertex Form
Learn to write, graph, and analyze quadratic functions in vertex form y = a(x - h)² + k.
Definition
- is the vertex of the parabola
- determines the width and direction of opening
- The axis of symmetry is the vertical line
- If : parabola opens upward (vertex is minimum)
- If : parabola opens downward (vertex is maximum)
- If : parabola is narrower than
- If : parabola is wider than
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Worked Examples
For , find the vertex, axis of symmetry, and direction of opening.
Identify from
means →
Identify as the constant
→
Write the vertex
Vertex = →
Find axis of symmetry
→
Determine direction from
→ Opens upward
Answer: Vertex: , Axis of symmetry: , Opens upward (narrower than )
Common Mistakes
Writing vertex as when equation has
Why it's wrong: When you see , this is , so , not .
Correct: Always rewrite to match : if you see , then .
Forgetting that affects width, not just direction
Why it's wrong: Students focus only on whether is positive or negative.
Correct: makes the parabola narrower; 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 to complete the square, you must also subtract to keep the equation balanced.
Interactive Visual
Quadratic Explorer
Controls width and direction
Shifts the parabola
Y-intercept
Vertex
(0, 0)
Axis of Symmetry
x = 0
Roots (x-intercepts)
x = 0 (double root)
Y-Intercept
(0, 0)
Direction
Opens upward
Discriminant
b² - 4ac = 0
Value Table
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
Interactive Sandbox
Interactive Grapher
Try these examples:
y = 2x + 1
m=2, b=1
Expression Calculator
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History
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Practice Problems
16 problemsWhat is the vertex of ?
Why It Matters
- 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 vertex is the focal point.
- Business: Profit functions are often quadratic. The vertex shows maximum profit.
- Architecture: Parabolic arches distribute weight efficiently. The vertex is the highest point.
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 feet. The vertex tells us the ball reaches 64 feet at seconds.
A water fountain shoots water following the path , where is height in meters and 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 :
Satellite Dishes
Parabolic dishes focus signals at the vertex. Engineers use vertex form to design the precise shape.
Example:
A dish with equation has its focal point above the vertex at the origin.
An engineer designs a parabolic reflector with equation . The reflector is 2 meters wide.
How deep is the dish at its edges (at meter)?
Step 1: Write the mathematical expression
Calculate when :
Key Takeaways
- 1Vertex form is where is the vertex
- 2The axis of symmetry is always
- 3: opens upward (minimum at vertex); : opens downward (maximum at vertex)
- 4: narrower parabola; : wider parabola
- 5To convert from standard form, complete the square
Frequently Asked Questions
Glossary
- Vertex
- The turning point of a parabola; the point in vertex form
- Axis of symmetry
- The vertical line that divides the parabola into two mirror images
- Vertex form
- The equation where 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
Standard vertex form equation
Vertex
Coordinates of the turning point
Axis of Symmetry
Vertical line through the vertex
Complete the Square
Convert to vertex form