Let's solve the transformation of point [tex]\( B(5,3) \)[/tex] step by step.
### Step 1: Reflection across the y-axis
First, we reflect point [tex]\( B(5,3) \)[/tex] across the y-axis. Reflecting across the y-axis changes the sign of the x-coordinate while the y-coordinate remains the same.
[tex]\[
B'(x', y') \quad \text{where} \quad x' = -x \quad \text{and} \quad y' = y
\][/tex]
So,
[tex]\[
B'(x', y') = (-5, 3)
\][/tex]
### Step 2: Dilation with center [tex]\((-1, 2)\)[/tex] and factor of 3
Next, we dilate [tex]\( B'(-5,3) \)[/tex] with the center of dilation at [tex]\((-1,2)\)[/tex] and a dilation factor of 3.
The formula for dilation with a specific center [tex]\((c_x, c_y)\)[/tex] and factor [tex]\( k \)[/tex] is:
[tex]\[
(x'', y'') = \left( c_x + k(x' - c_x) , c_y + k(y' - c_y) \right)
\][/tex]
Substituting the values [tex]\( c_x = -1 \)[/tex], [tex]\( c_y = 2 \)[/tex], [tex]\( x' = -5 \)[/tex], [tex]\( y' = 3 \)[/tex], and [tex]\( k = 3 \)[/tex], we get:
[tex]\[
x'' = -1 + 3(-5 + 1) = -1 + 3(-4) = -1 - 12 = -13
\][/tex]
[tex]\[
y'' = 2 + 3(3 - 2) = 2 + 3(1) = 2 + 3 = 5
\][/tex]
Hence,
[tex]\[
B''(-13, 5)
\][/tex]
Now, let's identify the correct option:
D. [tex]\( B''(-13, 5) \)[/tex] is the correct answer.
Thus, the transformation of [tex]\( B(5,3) \)[/tex] after reflection across the y-axis and dilation by a factor of 3 using the point [tex]\((-1,2)\)[/tex] as the center of dilation is [tex]\( \boxed{(-13,5)} \)[/tex].
