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I see that you have uploaded an image with several math exercises. I will help you solve these problems one by one. Let's begin by tackling each question in order.

### 1. **Determine the result of the transformation:**
   \[
   y = 2x + 6 \quad \text{shifted 8 units down.}
   \]
   **Solution:**
   Shifting the function down by 8 units affects the constant term:
   \[
   y = 2x + 6 - 8 = 2x - 2.
   \]

### 2. **Determine the result of translating point \( H = (2, -3) \) by the vector \( T = \begin{pmatrix} 4 \\ -2 \end{pmatrix} \).**

   **Solution:**
   Translation means adding the components of the vector \( T \) to the coordinates of point \( H \):
   \[
   H' = (2 + 4, -3 + (-2)) = (6, -5).
   \]

### 3. **The graph of \( y = x^2 + bx \) is shifted 2 units to the right. Determine the equation after this transformation.**

   **Solution:**
   A shift of 2 units to the right corresponds to replacing \( x \) with \( (x - 2) \) in the equation:
   \[
   y = (x - 2)^2 + b(x - 2).
   \]
   Expanding:
   \[
   y = (x^2 - 4x + 4) + b(x - 2) = x^2 + (b - 4)x + (4 - 2b).
   \]
   So, the transformed equation is:
   \[
   y = x^2 + (b - 4)x + (4 - 2b).
   \]

### 4. **Determine the reflection of point \( A(2, 3) \) with respect to various axes and lines:**
   - **a) With respect to the x-axis:**
     Reflection over the x-axis changes the sign of the y-coordinate:
     \[
     A' = (2, -3).
     \]
   - **b) With respect to the y-axis:**
     Reflection over the y-axis changes the sign of the x-coordinate:
     \[
     A' = (-2, 3).
     \]
   - **c) With respect to the line \( y = x \):**
     Swap the coordinates of the point:
     \[
     A' = (3, 2).
     \]
   - **d) With respect to the line \( y = -x \):**
     Swap and negate the coordinates:
     \[
     A' = (-3, -2).
     \]
   - **e) With respect to the origin (0, 0):**
     Negate both coordinates:
     \[
     A' = (-2, -3).
     \]
   - **f) With respect to the line \( x = 2 \):**
     The reflection across \( x = 2 \) means the x-coordinate is unchanged:
     \[
     A' = (2, 3).
     \]
   - **g) With respect to the line \( y = 1 \):**
     The reflection across \( y = 1 \) changes the y-coordinate according to the line:
     \[
     A' = (2, 2 \times 1 - 3) = (2, -1).
     \]

### 5. **Given \( f(x) = x^2 + x - 2 \), find the reflection of \( f(x) \):**
   - **a) Across the x-axis:**
     Reflection over the x-axis changes the sign of the function:
     \[
     f'(x) = -(x^2 + x - 2) = -x^2 - x + 2.
     \]
   - **b) Across the y-axis:**
     Reflection over the y-axis means replacing \( x \) with \( -x \):
     \[
     f'(x) = (-x)^2 + (-x) - 2 = x^2 - x - 2.
     \]

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Do you need detailed explanations for any of these steps or solutions? 

Here are five related questions to expand your understanding:
1. How do we perform a vertical stretch or compression on the function \( y = 2x + 6 \)?
2. How would the result change if the translation vector \( T \) in question 2 were \( T = \begin{pmatrix} -3 \\ 5 \end{pmatrix} \)?
3. What effect does shifting the graph of \( y = x^2 \) by 5 units to the left have on its equation?
4. How would the reflection of \( A(2, 3) \) look with respect to the line \( y = x + 1 \)?
5. What happens to a function \( f(x) \) if it undergoes a horizontal stretch by a factor of 2?

**Tip:** When reflecting a point or graph, always identify which coordinate or expression is changing to avoid confusion!

Math problems involving transformations and reflections in coordinate geometry.

This detailed guide covers how to perform transformations such as translations and reflections on functions and points in coordinate geometry. Learn how to shift graphs and reflect points across various axes and lines, complete with step-by-step solutions for high school students in Grades 10-12.

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