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The problem asks for the reflection of the function \( f(x) = x^2 - 4x + 3 \) with respect to:

a. The line \( x = -3 \)

b. The line \( y = -3 \)

Let's break it down step by step:

### a. Reflection with respect to \( x = -3 \):

To reflect a function across a vertical line like \( x = -3 \), we replace \( x \) in the function with \( -(x + 3) \). The reflection rule for vertical lines is \( x' = -(x + c) \) where \( c \) is the \( x \)-coordinate of the vertical line (here \( c = -3 \)).

1. Start with the original function \( f(x) = x^2 - 4x + 3 \).
2. Replace \( x \) with \( -(x + 3) \) to reflect across \( x = -3 \):

\[
f(-x - 3) = (-x - 3)^2 - 4(-x - 3) + 3
\]

Now, let's simplify this:

\[
= (x^2 + 6x + 9) + 4(x + 3) + 3
\]
\[
= x^2 + 6x + 9 + 4x + 12 + 3
\]
\[
= x^2 + 10x + 24
\]

So, the reflection of the function across \( x = -3 \) is \( f(x) = x^2 + 10x + 24 \).

### b. Reflection with respect to \( y = -3 \):

To reflect a function across a horizontal line like \( y = -3 \), we apply the following rule: the reflection is \( f'(x) = -f(x) - 6 \), where 6 comes from doubling the distance between the function's original vertex and the line of reflection (since the line is at \( y = -3 \)).

1. The original function is \( f(x) = x^2 - 4x + 3 \).
2. To reflect across \( y = -3 \), we first negate the function and subtract 6:

\[
f'(x) = -(x^2 - 4x + 3) - 6
\]
\[
= -x^2 + 4x - 3 - 6
\]
\[
= -x^2 + 4x - 9
\]

So, the reflection of the function across \( y = -3 \) is \( f(x) = -x^2 + 4x - 9 \).

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Would you like more details on any step, or have further questions?

Here are 5 related questions to consider:
1. How do you reflect a function across the y-axis?
2. What happens to the vertex of a parabola during reflection over a line?
3. How can you find the vertex of a quadratic function?
4. What is the general rule for reflecting a graph across a vertical line?
5. How can transformations affect the symmetry of a function?

**Tip:** Always double-check the signs when reflecting functions to avoid calculation mistakes!

Tentukan persamaan bayangan dari fungsi f(x) = x^2 - 4x + 3 jika di refleksikan terhadap: a. Garis x = -3, b. Garis y = -3

Discover how to reflect the quadratic function f(x) = x^2 - 4x + 3 across the vertical line x = -3 and the horizontal line y = -3. This problem covers key concepts in quadratic transformations and function reflections, suitable for students in Grades 10-12.

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