Question

The quadratic parent function has been reflected down, stretched vertically by a factor of 1/3, shifted vertically by 9 units and shifted horizontally by -12 units.
1a) write the function g(x) in vertex form

1b) then rewrite the vertex form into standard form

Answer 1

Answer:

1a) f(x) = -1/3(x + 12)² + 9

1b) f(x) = -1/3x² - 8x - 39

General Formulas and Concepts:

Algebra I

Equality Properties

Expand by FOIL (First Outside Inside Last)

Standard Form: ax² + bx + c = 0

Transformations Graph: f(x) = a(bx - c)² + d

• a = vertical shrink/stretch

• b = horizontal shrink/stretch

• c = horizontal movement left/right

• d = vertical movement up/down

Step-by-step explanation:

Step 1: Define

Reflected down and vertically stretched by 1/3:   a = -1/3

Shifted vertically by 9 units:   d = 9

Shifted horizontally by -12 units:   c = -12

Step 2: Write Vertex Form

1. Define:                    f(x) = a(bx - c)² + d
2. Substitute:              f(x) = -1/3(x + 12)² + 9

Step 3: Write Standard Form

1. Define:                              f(x) = -1/3(x + 12)² + 9
2. Expand:                             f(x) = -1/3(x² + 24x + 144) + 9
3. Distribute -1/3:                  f(x) = -1/3x² - 8x - 48 + 9
4. Combine like terms:         f(x) = -1/3x² - 8x - 39

And we have our final answers!