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 $f(x) = -x^2 + 4x - 9$.

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!