https://file.aiquickdraw.com/mathbot/file/17278904487091x3wsby6.jpg

Give me every partner function and the way it moves

Let's break down each function given in the image. I will describe the transformations applied to each parent function \( f(x) \), as well as how the graph of \( g(x) \) is transformed from its parent function.

### 24. \( g(x) = 3|x| - 4 \)
- **Parent function**: \( f(x) = |x| \) (Absolute value function)
- **Transformations**:
  1. Vertical stretch by a factor of 3 (the 3 in front of \( |x| \)).
  2. Vertical shift downward by 4 units (the \(-4\)).

### 25. \( g(x) = 3\sqrt{x} + 8 \)
- **Parent function**: \( f(x) = \sqrt{x} \) (Square root function)
- **Transformations**:
  1. Vertical stretch by a factor of 3 (the 3 in front of \( \sqrt{x} \)).
  2. Vertical shift upward by 8 units (the \( +8 \)).

### 26. \( g(x) = \frac{4}{x+1} \)
- **Parent function**: \( f(x) = \frac{1}{x} \) (Reciprocal function)
- **Transformations**:
  1. Horizontal shift left by 1 unit (the \( +1 \) inside the denominator).
  2. Vertical stretch by a factor of 4 (the 4 in the numerator).

### 27. \( g(x) = 2|x - 6| \)
- **Parent function**: \( f(x) = |x| \)
- **Transformations**:
  1. Horizontal shift right by 6 units (the \(-6\) inside the absolute value).
  2. Vertical stretch by a factor of 2 (the 2 in front of \( |x - 6| \)).

### 28. \( g(x) = -5|x - 2| \)
- **Parent function**: \( f(x) = |x| \)
- **Transformations**:
  1. Horizontal shift right by 2 units (the \(-2\) inside the absolute value).
  2. Vertical stretch by a factor of 5.
  3. Reflection across the x-axis (because of the negative sign in front of the 5).

### 29. \( g(x) = -2|x + 5| \)
- **Parent function**: \( f(x) = |x| \)
- **Transformations**:
  1. Horizontal shift left by 5 units (the \( +5 \) inside the absolute value).
  2. Vertical stretch by a factor of 2.
  3. Reflection across the x-axis (the negative sign in front of 2).

### 30. \( g(x) = \frac{1}{6x} + 7 \)
- **Parent function**: \( f(x) = \frac{1}{x} \)
- **Transformations**:
  1. Horizontal compression by a factor of \( 1/6 \) (the \( 6x \) in the denominator).
  2. Vertical shift upward by 7 units (the \( +7 \)).

### 31. \( g(x) = \frac{\sqrt{x + 3}}{4} \)
- **Parent function**: \( f(x) = \sqrt{x} \)
- **Transformations**:
  1. Horizontal shift left by 3 units (the \( +3 \) inside the square root).
  2. Vertical compression by a factor of 4 (the division by 4).

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Would you like further details on any of the transformations, or do you have any questions about these functions?

Here are 5 additional related questions to expand on this topic:
1. What is the parent function of \( g(x) = -|x| + 3 \), and how does it move?
2. How would the graph of \( g(x) = 2\sqrt{x - 4} - 1 \) be transformed?
3. What happens if the coefficient inside an absolute value function is negative, like in \( g(x) = -|x - 2| + 5 \)?
4. How does horizontal stretching or compression affect the graph of a function?
5. What are the key differences between vertical and horizontal transformations?

**Tip**: Always remember that vertical transformations affect the output (y-values), while horizontal transformations affect the input (x-values).

Explore transformations of various functions including absolute value, square root, and reciprocal functions. Learn how vertical stretches, reflections, and horizontal shifts affect each graph. Suitable for high school math students.

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