Algebra worksheet identifying function transformations with a riddle.

Problem Analysis:

The task involves identifying the transformation applied to each given function relative to its parent function. Once the transformations are identified, the corresponding letters will spell out the answer to the riddle: "What can go up a chimney down, but can’t go down a chimney up?"

Step-by-Step Solution:

#### 1. Identify the Parent Functions and Transformations:
- Parent functions are the simplest forms of each type of function (e.g., \( f(x) = x^2 \), \( f(x) = \sqrt{x} \), etc.).
- Transformations include shifts, reflections, stretches, and shrinks.

#### 2. Analyze Each Function:

##### Function B: \( f(x) = (x - 3)^2 \)
- Parent function: \( f(x) = x^2 \)
- Transformation: The term \( (x - 3) \) indicates a horizontal shift to the right by 3 units.
- Match: Shift right
- Letter: B

##### Function A: \( f(x) = \sqrt{x} - 4 \)
- Parent function: \( f(x) = \sqrt{x} \)
- Transformation: The term \( -4 \) indicates a vertical shift downward by 4 units.
- Match: Shift down
- Letter: A

##### Function L: \( f(x) = \frac{1}{2}x^3 \)
- Parent function: \( f(x) = x^3 \)
- Transformation: The coefficient \( \frac{1}{2} \) indicates a vertical shrink by a factor of \( \frac{1}{2} \).
- Match: Shrink
- Letter: L

##### Function E: \( f(x) = |x + 6| \)
- Parent function: \( f(x) = |x| \)
- Transformation: The term \( x + 6 \) indicates a horizontal shift to the left by 6 units.
- Match: Shift left
- Letter: E

##### Function U: \( f(x) = -2^x \)
- Parent function: \( f(x) = 2^x \)
- Transformation: The negative sign indicates a reflection across the x-axis.
- Match: Reflect x-axis
- Letter: U

##### Function R: \( f(x) = 0.25|x| \)
- Parent function: \( f(x) = |x| \)
- Transformation: The coefficient \( 0.25 \) indicates a vertical shrink by a factor of \( 0.25 \).
- Match: Shrink
- Letter: R

##### Function N: \( f(x) = x^2 + 1 \)
- Parent function: \( f(x) = x^2 \)
- Transformation: The term \( +1 \) indicates a vertical shift upward by 1 unit.
- Match: Shift up
- Letter: N

##### Function M: \( f(x) = 0.5^x \)
- Parent function: \( f(x) = 2^x \) (Note: \( 0.5^x = (2^{-1})^x = 2^{-x} \))
- Transformation: This is equivalent to reflecting \( f(x) = 2^x \) across the y-axis.
- Match: Reflect y-axis
- Letter: M

#### 3. Match Letters to Transforms:
- Shift down: A
- Shift up: N
- Reflect x-axis: U
- Reflect y-axis: M
- Shift right: B
- Shrink: L, R
- Stretch: None (not applicable here)
- Shift down: A (repeated)

#### 4. Spell Out the Answer:
- The letters corresponding to the transformations are: A, N, U, M, B, L, R
- Arranging these letters spells out: ANUMBRL

#### 5. Solve the Riddle:
- The riddle "What can go up a chimney down, but can’t go down a chimney up?" is a classic one.
- The answer is: ASH

Final Answer:

\[
\boxed{\text{ASH}}
\]