Parabolas
📌 What is a Parabola?
A parabola is a U-shaped curve that is the graph of a quadratic function. It is also defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).
Forms of Parabola Equations
Three Main Forms:
1. Vertex Form (Graphing Form):
\( y = a(x - h)^2 + k \) (opens up/down)
Vertex: \( (h, k) \)
2. Standard Form (General Form):
\( y = ax^2 + bx + c \) (opens up/down)
3. Conic Form (Focus-Directrix Form):
\( (x - h)^2 = 4p(y - k) \) (opens up/down)
Direction a Parabola Opens
For Vertical Parabolas: \( y = a(x - h)^2 + k \)
- If \( a > 0 \): Parabola opens UPWARD (U-shape)
- If \( a < 0 \): Parabola opens DOWNWARD (∩-shape)
- The larger \( |a| \), the narrower the parabola
- The smaller \( |a| \), the wider the parabola
For Horizontal Parabolas: \( x = a(y - k)^2 + h \)
- If \( a > 0 \): Parabola opens RIGHT (→)
- If \( a < 0 \): Parabola opens LEFT (←)
📝 Examples - Direction:
Example 1: \( y = 3(x - 2)^2 + 1 \)
\( a = 3 > 0 \) → Opens UPWARD
Example 2: \( y = -2x^2 + 4x - 1 \)
\( a = -2 < 0 \) → Opens DOWNWARD
Example 3: \( x = (y + 1)^2 - 3 \)
\( a = 1 > 0 \) → Opens RIGHT
Finding the Vertex of a Parabola
Method 1: From Vertex Form
If the equation is \( y = a(x - h)^2 + k \):
Vertex = \( (h, k) \)
Read directly from the equation!
Method 2: From Standard Form
If the equation is \( y = ax^2 + bx + c \):
Step 1: Find x-coordinate of vertex:
\( x = -\frac{b}{2a} \)
Step 2: Substitute x back to find y-coordinate:
\( y = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c \)
📝 Examples - Finding Vertex:
Example 1: Find vertex of \( y = 2(x - 3)^2 + 5 \)
From vertex form: \( h = 3, k = 5 \)
Vertex: \( (3, 5) \)
Example 2: Find vertex of \( y = x^2 - 6x + 11 \)
\( a = 1, b = -6, c = 11 \)
\( x = -\frac{-6}{2(1)} = \frac{6}{2} = 3 \)
\( y = (3)^2 - 6(3) + 11 = 9 - 18 + 11 = 2 \)
Vertex: \( (3, 2) \)
Example 3: Find vertex of \( y = -2x^2 + 8x - 3 \)
\( x = -\frac{8}{2(-2)} = -\frac{8}{-4} = 2 \)
\( y = -2(2)^2 + 8(2) - 3 = -8 + 16 - 3 = 5 \)
Vertex: \( (2, 5) \)
Axis of Symmetry
Definition:
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 Parabolas (opens up/down):
\( x = h \) (vertical line through vertex)
In standard form: \( x = -\frac{b}{2a} \)
For Horizontal Parabolas (opens left/right):
\( y = k \) (horizontal line through vertex)
📝 Examples - Axis of Symmetry:
Example 1: \( y = (x + 4)^2 - 7 \)
Vertex: \( (-4, -7) \) → Axis of symmetry: \( x = -4 \)
Example 2: \( y = 3x^2 + 12x + 5 \)
\( x = -\frac{12}{2(3)} = -\frac{12}{6} = -2 \)
Axis of symmetry: \( x = -2 \)
Focus and Directrix
Definitions:
Focus:
A fixed point inside the parabola such that every point on the parabola is equidistant from the focus and the directrix.
Directrix:
A fixed line outside the parabola perpendicular to the axis of symmetry.
Formulas for Vertical Parabolas:
Standard form: \( (x - h)^2 = 4p(y - k) \) where vertex is \( (h, k) \)
Focus:
\( (h, k + p) \)
Directrix:
\( y = k - p \)
Where:
- \( p > 0 \): parabola opens upward
- \( p < 0 \): parabola opens downward
- \( |p| \) = distance from vertex to focus (and from vertex to directrix)
Converting Between Forms:
From vertex form \( y = a(x - h)^2 + k \) to conic form:
\( p = \frac{1}{4a} \)
📝 Examples - Focus and Directrix:
Example 1: Find focus and directrix of \( y = \frac{1}{8}(x - 2)^2 + 3 \)
Vertex: \( (2, 3) \)
\( a = \frac{1}{8} \), so \( p = \frac{1}{4 \cdot \frac{1}{8}} = \frac{1}{\frac{1}{2}} = 2 \)
Focus: \( (2, 3 + 2) = (2, 5) \)
Directrix: \( y = 3 - 2 = 1 \)
Example 2: Given \( (x - 1)^2 = 12(y + 2) \), find focus and directrix
This is in form \( (x - h)^2 = 4p(y - k) \)
Vertex: \( (1, -2) \)
\( 4p = 12 \), so \( p = 3 \)
Focus: \( (1, -2 + 3) = (1, 1) \)
Directrix: \( y = -2 - 3 = -5 \)
Writing Equations in Vertex Form
From a Graph:
- Identify the vertex \( (h, k) \) from the graph
- Choose another point \( (x, y) \) on the parabola
- Write \( y = a(x - h)^2 + k \)
- Substitute the point to find \( a \)
- Write the final equation with the value of \( a \)
📝 Example - From Graph:
A parabola has vertex at \( (3, -2) \) and passes through \( (5, 6) \). Find the equation.
Step 1: Vertex form: \( y = a(x - 3)^2 - 2 \)
Step 2: Substitute \( (5, 6) \):
\( 6 = a(5 - 3)^2 - 2 \)
\( 6 = a(2)^2 - 2 \)
\( 6 = 4a - 2 \)
\( 8 = 4a \)
\( a = 2 \)
Equation: \( y = 2(x - 3)^2 - 2 \)
Using Properties (Focus/Directrix):
If given focus and directrix, or focus and vertex:
- Find the vertex (midpoint between focus and directrix)
- Find \( p \) (distance from vertex to focus)
- Calculate \( a = \frac{1}{4p} \)
- Write equation: \( y = a(x - h)^2 + k \)
Converting from Standard to Vertex Form
Method: Completing the Square
To convert \( y = ax^2 + bx + c \) to vertex form:
- Factor out \( a \) from the first two terms
- Take half of the \( x \)-coefficient inside parentheses, square it
- Add and subtract this value inside the parentheses
- Factor the perfect square trinomial
- Simplify to get \( y = a(x - h)^2 + k \)
📝 Example - Completing the Square:
Convert \( y = 2x^2 + 12x + 13 \) to vertex form
Step 1: Factor out \( a = 2 \):
\( y = 2(x^2 + 6x) + 13 \)
Step 2: Half of 6 is 3, squared is 9:
\( y = 2(x^2 + 6x + 9 - 9) + 13 \)
Step 3: Factor perfect square:
\( y = 2[(x + 3)^2 - 9] + 13 \)
Step 4: Distribute and simplify:
\( y = 2(x + 3)^2 - 18 + 13 \)
\( y = 2(x + 3)^2 - 5 \)
Vertex: \( (-3, -5) \)
📝 Example 2 - More Practice:
Convert \( y = -x^2 + 4x + 1 \) to vertex form
Factor out -1:
\( y = -(x^2 - 4x) + 1 \)
Complete the square (half of -4 is -2, squared is 4):
\( y = -(x^2 - 4x + 4 - 4) + 1 \)
\( y = -[(x - 2)^2 - 4] + 1 \)
\( y = -(x - 2)^2 + 4 + 1 \)
\( y = -(x - 2)^2 + 5 \)
Vertex: \( (2, 5) \)
Finding Properties from General Form
Given \( y = ax^2 + bx + c \), find:
1. Direction:
\( a > 0 \) → opens up; \( a < 0 \) → opens down
2. Vertex:
\( x = -\frac{b}{2a} \), then find \( y \) by substituting
3. Axis of Symmetry:
\( x = -\frac{b}{2a} \)
4. Y-intercept:
\( (0, c) \)
5. X-intercepts (if they exist):
Solve \( ax^2 + bx + c = 0 \) using factoring, quadratic formula, etc.
Graphing Parabolas
Step-by-Step Graphing:
- Find the vertex and plot it
- Determine the direction (up/down from sign of \( a \))
- Draw the axis of symmetry (dashed line through vertex)
- Find the y-intercept and its symmetric point
- Find x-intercepts if they exist (solve equation = 0)
- Plot additional points on one side and use symmetry
- Sketch the smooth curve through all points
📝 Example - Complete Graphing:
Graph \( y = -x^2 + 4x - 3 \)
Step 1: Vertex
\( x = -\frac{4}{2(-1)} = 2 \)
\( y = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1 \)
Vertex: \( (2, 1) \)
Step 2: Direction: \( a = -1 < 0 \) → opens DOWN
Step 3: Axis of symmetry: \( x = 2 \)
Step 4: Y-intercept: \( (0, -3) \)
Symmetric point: \( (4, -3) \)
Step 5: X-intercepts: Solve \( -x^2 + 4x - 3 = 0 \)
\( x^2 - 4x + 3 = 0 \)
\( (x - 1)(x - 3) = 0 \)
X-intercepts: \( (1, 0) \) and \( (3, 0) \)
Step 6: Sketch parabola opening down through these points
⚡ Quick Summary
| Property | Formula/Rule |
|---|---|
| Vertex Form | \( y = a(x-h)^2 + k \) |
| Standard Form | \( y = ax^2 + bx + c \) |
| Vertex | \( (h, k) \) or \( (-\frac{b}{2a}, y) \) |
| Axis of Symmetry | \( x = h \) or \( x = -\frac{b}{2a} \) |
| Direction | \( a > 0 \): up; \( a < 0 \): down |
| Focus (vertical) | \( (h, k+p) \) |
| Directrix (vertical) | \( y = k - p \) |
| Value of p | \( p = \frac{1}{4a} \) |
📚 Key Points to Remember
- Vertex form makes it easy to identify vertex and direction
- Use completing the square to convert standard to vertex form
- Axis of symmetry always passes through the vertex
- Focus is inside the parabola, directrix is outside
- The closer the focus to the vertex, the narrower the parabola
- All points on a parabola are equidistant from focus and directrix
- Y-intercept is always at \( (0, c) \) in standard form
- Use symmetry to find corresponding points when graphing