Standard Form (General Form):

\[ f(x) = ax^2 + bx + c \]

where \( a, b, c \) are constants and \( a \neq 0 \)

• \( c \) is the y-intercept

Vertex Form:

\[ f(x) = a(x - h)^2 + k \]

where \( (h, k) \) is the vertex

• \( h \) = horizontal shift, \( k \) = vertical shift

Factored Form (Intercept Form):

\[ f(x) = a(x - p)(x - q) \]

where \( p \) and \( q \) are x-intercepts (zeros)

Graph Shape: Parabola (U-shaped curve)

Direction:

• If \( a > 0 \): Opens UPWARD (minimum vertex)

• If \( a < 0 \): Opens DOWNWARD (maximum vertex)

Vertex:

• The turning point (highest or lowest point)

• From standard form: \( x = -\frac{b}{2a} \), then find \( y \)

Axis of Symmetry:

\[ x = -\frac{b}{2a} \quad \text{or} \quad x = h \]

Vertical line through the vertex that divides parabola in half

\[ \text{Domain: } (-\infty, \infty) \text{ or } \mathbb{R} \]

All real numbers - quadratic functions are defined for all x-values

Range depends on the vertex and direction of opening:

If parabola opens UPWARD (\( a > 0 \)):

\[ \text{Range: } [k, \infty) \]

where \( k \) is the y-coordinate of the vertex (minimum value)

If parabola opens DOWNWARD (\( a < 0 \)):

\[ \text{Range: } (-\infty, k] \]

where \( k \) is the y-coordinate of the vertex (maximum value)

\[ f(x) = x^2 \]

Vertex at origin (0, 0), opens upward

\[ f(x) = a(x - h)^2 + k \]

Step 1: Find the vertex

• From vertex form: \( (h, k) \)

• From standard form: \( x = -\frac{b}{2a} \), then find \( y \)

Step 2: Find the axis of symmetry

Vertical line: \( x = h \) or \( x = -\frac{b}{2a} \)

Step 3: Find y-intercept

Set \( x = 0 \) and solve for \( y \) (this is \( c \) in standard form)

Step 4: Find x-intercepts (if any)

Set \( y = 0 \) and solve for \( x \)

Use factoring, completing the square, or quadratic formula

Step 5: Plot additional points

Choose x-values on both sides of axis of symmetry

Step 6: Draw the parabola

Connect points with smooth U-shaped curve

Convert \( f(x) = ax^2 + bx + c \) to \( f(x) = a(x - h)^2 + k \)

Steps:

1. Factor out \( a \) from first two terms (if \( a \neq 1 \))

2. Complete the square inside parentheses

3. Balance the equation

4. Simplify to vertex form

Example:

Convert \( f(x) = 2x^2 - 12x + 5 \) to vertex form

Step 1: \( f(x) = 2(x^2 - 6x) + 5 \)

Step 2: \( f(x) = 2(x^2 - 6x + 9 - 9) + 5 \)

Step 3: \( f(x) = 2(x^2 - 6x + 9) - 18 + 5 \)

Answer: \( f(x) = 2(x - 3)^2 - 13 \)

Quick Formula:

• \( h = -\frac{b}{2a} \)

• \( k = f(h) \) (substitute h back into function)

• Write: \( f(x) = a(x - h)^2 + k \)

Formula:

\[ f(x) = a(x - p)(x - q) \]

where \( p \) and \( q \) are the zeros

Example 1: Given zeros only

Zeros: \( x = 3 \) and \( x = -5 \)

Function: \( f(x) = (x - 3)(x + 5) \) or \( f(x) = x^2 + 2x - 15 \)

Example 2: Given zeros and another point

Zeros: \( x = 2 \) and \( x = 6 \); Point: (0, -24)

Step 1: \( f(x) = a(x - 2)(x - 6) \)

Step 2: Use point (0, -24): \( -24 = a(0 - 2)(0 - 6) \)

Step 3: \( -24 = a(12) \), so \( a = -2 \)

Answer: \( f(x) = -2(x - 2)(x - 6) \)

Step 1: Start with vertex form

\( f(x) = a(x - h)^2 + k \) where \( (h, k) \) is the vertex

Step 2: Substitute the other point

Use the coordinates to solve for \( a \)

Step 3: Write final equation

Example:

Vertex: (3, -4); Point: (5, 8)

Step 1: \( f(x) = a(x - 3)^2 - 4 \)

Step 2: Substitute (5, 8): \( 8 = a(5 - 3)^2 - 4 \)

Step 3: \( 8 = 4a - 4 \), so \( 12 = 4a \), thus \( a = 3 \)

Answer: \( f(x) = 3(x - 3)^2 - 4 \)

Step 1: Use standard form \( f(x) = ax^2 + bx + c \)

Step 2: Substitute each point to create 3 equations

Step 3: Solve the system for a, b, and c

Example:

Points: (1, 4), (2, 1), (3, 2)

Answer: \( f(x) = 2x^2 - 7x + 9 \)

Substitute each x-value into the quadratic function and calculate y-values

Example:

Complete table for \( f(x) = x^2 - 4x + 3 \)

Note: Vertex is at x = 2 (axis of symmetry), values are symmetric around it