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This image contains a set of problems dealing with transformations, reflections, and dilations of functions.

### 1. **Translations**
You are given functions and translation vectors. To apply a translation to a function, the following rules apply:
- Horizontal translation \( f(x + c) \) shifts the function by \( c \) units in the opposite direction of the sign.
- Vertical translation \( f(x) + d \) shifts the function by \( d \) units up or down.

#### a) \( f(x) = 3x^2 - x - 2 \) and translation \( [-1, -2] \):
- The horizontal translation by \(-1\): Replace \(x\) with \(x + 1\).
- The vertical translation by \(-2\): Subtract 2 from the function.
  \[
  g(x) = f(x + 1) - 2
  \]
  First, compute \( f(x + 1) \):
  \[
  f(x + 1) = 3(x + 1)^2 - (x + 1) - 2 = 3(x^2 + 2x + 1) - x - 1 - 2 = 3x^2 + 6x + 3 - x - 3 - 2 = 3x^2 + 5x - 2
  \]
  Now apply the vertical translation:
  \[
  g(x) = 3x^2 + 5x - 2 - 2 = 3x^2 + 5x - 4
  \]

#### b) \( f(x) = 3x^2 - x - 2 \) and translation \( [-4, -1] \):
- Horizontal translation by \(-4\): Replace \(x\) with \(x + 4\).
- Vertical translation by \(-1\): Subtract 1 from the function.
  \[
  g(x) = f(x + 4) - 1
  \]
  First, compute \( f(x + 4) \):
  \[
  f(x + 4) = 3(x + 4)^2 - (x + 4) - 2 = 3(x^2 + 8x + 16) - x - 4 - 2 = 3x^2 + 24x + 48 - x - 6 = 3x^2 + 23x + 42
  \]
  Now apply the vertical translation:
  \[
  g(x) = 3x^2 + 23x + 42 - 1 = 3x^2 + 23x + 41
  \]

### 2. **Reflections**
Reflections can occur over the x-axis, y-axis, or other lines.

#### a) \( f(x) = 3x^2 + 1 \) and reflection over \(x = -2\):
For a reflection over \(x = -2\), the transformation involves shifting \(x\) and reflecting. Replace \(x\) with \(-x - 4\).
  \[
  g(x) = f(-x - 4) = 3(-x - 4)^2 + 1 = 3(x^2 + 8x + 16) + 1 = 3x^2 + 24x + 48 + 1 = 3x^2 + 24x + 49
  \]

#### b) \( f(x) = 4x + 6 \) and reflection over \(y = -x\):
To reflect over the line \(y = -x\), swap \(x\) with \(-y\) in the equation.
  \[
  g(x) = -f(-x) = -(4(-x) + 6) = -(-4x + 6) = 4x - 6
  \]

### 3. **Dilations**
Dilation involves scaling the function vertically or horizontally.

#### a) \( f(x) = 6x^2 - 4x + 2 \) and vertical scaling by \( \frac{1}{2} \):
To vertically scale the function, multiply the entire function by \( \frac{1}{2} \):
  \[
  g(x) = \frac{1}{2}f(x) = \frac{1}{2}(6x^2 - 4x + 2) = 3x^2 - 2x + 1
  \]

#### b) \( f(x) = 2x^2 + 3x - 4 \) and horizontal scaling by 2:
For horizontal scaling, replace \(x\) with \( \frac{x}{2} \) in the function:
  \[
  g(x) = f\left(\frac{x}{2}\right) = 2\left(\frac{x}{2}\right)^2 + 3\left(\frac{x}{2}\right) - 4 = 2\left(\frac{x^2}{4}\right) + 3\left(\frac{x}{2}\right) - 4 = \frac{x^2}{2} + \frac{3x}{2} - 4
  \]

#### c) \( f(x) = ? \) and vertical scaling by 2 for \( g(x) = 4x - 2 \):
To get \( f(x) \) from \( g(x) = 4x - 2 \) and vertical scaling by 2, divide the function by 2:
  \[
  f(x) = \frac{1}{2}(4x - 2) = 2x - 1
  \]

Let me know if you need further details or explanation.

### 5 Related Questions:
1. How do you apply horizontal scaling in function transformations?
2. What changes occur to a function when it is reflected over the y-axis?
3. Can you explain how to reflect a function over a vertical line other than the y-axis?
4. How does vertical dilation affect the graph of a quadratic function?
5. What is the difference between horizontal and vertical translations in terms of function graphs?

**Tip**: When transforming functions, remember that horizontal transformations (translations and dilations) affect the input \(x\), while vertical transformations affect the output of the function.

The image contains problems related to transformations of functions, including translations, reflections, and dilations.

Learn how to apply translations, reflections, and dilations to functions. This page provides step-by-step solutions to problems involving quadratic functions, including vertical and horizontal transformations. Suitable for high school students in Grades 9-11.

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