To solve this problem, we need to find the equation of a parabola after it has been shifted. The original equation of the parabola is given as [tex]\( y = 3(x + 5)^2 - 2 \)[/tex], and the vertex is at [tex]\((-5, -2)\)[/tex].
The steps to solve the problem:
1. Determine the new vertex after the shift:
- The parabola is shifted 6 units to the right.
- The [tex]\( x \)[/tex]-coordinate of the vertex will increase by 6 units: [tex]\(-5 + 6 = 1\)[/tex].
- The parabola is also shifted 1 unit down.
- The [tex]\( y \)[/tex]-coordinate of the vertex will decrease by 1 unit: [tex]\(-2 - 1 = -3\)[/tex].
2. Write the equation with the new vertex:
- The general form of a parabola's equation is [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex.
- With the vertex now at [tex]\((1, -3)\)[/tex] and the same value for [tex]\( a = 3 \)[/tex], we can substitute [tex]\( h = 1 \)[/tex] and [tex]\( k = -3 \)[/tex] into the general form:
[tex]\[
y = 3(x - 1)^2 - 3
\][/tex]
3. Compare with the options given:
- Option A: [tex]\( y = 3(x - 1)^2 - 1 \)[/tex] (Incorrect: [tex]\(-1 \neq -3\)[/tex])
- Option B: [tex]\( y = 3(x + 11)^2 - 1 \)[/tex] (Incorrect: The vertex would need to be at [tex]\(-11\)[/tex], not [tex]\(1\)[/tex], and [tex]\( -1 \neq -3\)[/tex])
- Option C: [tex]\( y = 3(x - 1)^2 - 3 \)[/tex] (Correct: Matches exactly)
- Option D: [tex]\( y = 3(x + 11)^2 - 3 \)[/tex] (Incorrect: The vertex would need to be at [tex]\(-11\)[/tex], not [tex]\(1\)[/tex])
Hence, the correct answer is Option C: [tex]\( y = 3(x - 1)^2 - 3 \)[/tex].
