Vertical Parabola (Opens Up/Down)

y = a(x - h)² + k

(h, k) = Vertex (the turning point)

a = Controls width and direction of opening

• If a > 0 → Opens UPWARD (U-shape)

• If a < 0 → Opens DOWNWARD (∩-shape)

• |a| > 1 → Narrower (steeper)

• |a| < 1 → Wider (flatter)

Horizontal Parabola (Opens Left/Right)

x = a(y - k)² + h

(h, k) = Vertex

• If a > 0 → Opens RIGHT

• If a < 0 → Opens LEFT

Vertical Parabola

y = ax² + bx + c

a, b, c = Constants (a ≠ 0)

Horizontal Parabola

x = ay² + by + c

From Vertex Form

Given: y = a(x - h)² + k

Vertex = (h, k)

⚠️ Note: y = (x - 3)² + 5 → Vertex is (3, 5), not (-3, 5)

From Standard Form

Given: y = ax² + bx + c

h = -b/(2a)

k = f(h) = a(h)² + b(h) + c

Vertex = (-b/(2a), f(-b/(2a)))

📝 Example

Find the vertex of: y = 2x² - 8x + 3

Solution:

• a = 2, b = -8, c = 3

• h = -(-8)/(2×2) = 8/4 = 2

• k = 2(2)² - 8(2) + 3 = 8 - 16 + 3 = -5

Vertex: (2, -5)

What is the Axis of Symmetry?

The axis of symmetry is a vertical (or horizontal) line that divides the parabola into two mirror-image halves. It always passes through the vertex.

For Vertical Parabola

From Vertex Form: y = a(x - h)² + k

Axis of Symmetry: x = h

From Standard Form: y = ax² + bx + c

Axis of Symmetry: x = -b/(2a)

For Horizontal Parabola

From Vertex Form: x = a(y - k)² + h

Axis of Symmetry: y = k

Definitions

• Focus: A fixed point inside the parabola

• Directrix: A fixed line outside the parabola

A parabola is the set of all points equidistant from the focus and directrix.

For Vertical Parabola: y = a(x - h)² + k

Distance from vertex to focus:

p = 1/(4a)

Focus:

F = (h, k + 1/(4a))

Directrix:

y = k - 1/(4a)

For Horizontal Parabola: x = a(y - k)² + h

Focus:

F = (h + 1/(4a), k)

Directrix:

x = h - 1/(4a)

📝 Example

Find focus and directrix of: y = (x - 2)² + 1

Solution:

• Vertex: (h, k) = (2, 1)

• a = 1

• 1/(4a) = 1/(4×1) = 1/4

• Focus: (2, 1 + 1/4) = (2, 5/4) or (2, 1.25)

• Directrix: y = 1 - 1/4 = 3/4 or 0.75

Method: Completing the Square

Steps:

Step 1: If a ≠ 1, factor out 'a' from the x² and x terms

Step 2: Take half of the x coefficient (inside parentheses), square it

Step 3: Add and subtract this value inside the parentheses

Step 4: Factor the perfect square trinomial

Step 5: Simplify to get vertex form

💡 Completing the Square Formula

For x² + bx, add (b/2)²:

x² + bx + (b/2)² = (x + b/2)²

📝 Example

Convert to vertex form: y = 2x² + 12x + 10

Solution:

Step 1: Factor out 2: y = 2(x² + 6x) + 10

Step 2: Half of 6 is 3, and 3² = 9

Step 3: Add and subtract 9 inside: y = 2(x² + 6x + 9 - 9) + 10

Step 4: Factor: y = 2[(x + 3)² - 9] + 10

Step 5: Simplify: y = 2(x + 3)² - 18 + 10

Answer: y = 2(x + 3)² - 8

Vertex: (-3, -8)

Method 1: Given Vertex and One Point

Step 1: Write y = a(x - h)² + k using the vertex (h, k)

Step 2: Substitute the given point (x, y) into the equation

Step 3: Solve for 'a' and write the final equation

📝 Example 1

Write equation with vertex (3, -2) passing through (5, 6).

Solution:

Step 1: y = a(x - 3)² - 2

Step 2: Substitute (5, 6): 6 = a(5 - 3)² - 2

Step 3: 6 = a(4) - 2 → 8 = 4a → a = 2

Equation: y = 2(x - 3)² - 2

Method 2: Given Focus and Directrix

Step 1: Find the vertex (midpoint between focus and directrix)

Step 2: Find the distance p from vertex to focus (or directrix)

Step 3: Calculate a = 1/(4p)

Step 4: Write equation: y = a(x - h)² + k

From Vertex Form

Steps:

Step 1: Identify the vertex (h, k) and plot it

Step 2: Determine direction of opening from sign of 'a'

Step 3: Draw the axis of symmetry (vertical or horizontal line through vertex)

Step 4: Find additional points by substituting x values (or y values)

Step 5: Use symmetry to plot mirror points and sketch the parabola

From Standard Form

Steps:

Step 1: Find the vertex using h = -b/(2a) and k = f(h)

Step 2: Plot the vertex

Step 3: Find y-intercept (set x = 0): y = c

Step 4: Find x-intercepts (if any) by setting y = 0 and solving

Step 5: Sketch the parabola through all points

🔹 Parabola Equation Forms

🔹 Key Formulas