Question

The vertex of the parabola $2y = -3x^2 + 48x - 200$, is

• A. $(8, -8)$
• B. $(8, -6)$
• C. $(-4, 8)$
• D. $(8, -4)$
• E. $(-4, -8)$

Hint:

Differentiating the expression \( f(x) = -3x^2 + 48x - 200 \) and setting it to zero is often the fastest way to find the $x$-coordinate of the vertex: \( -6x + 48 = 0 \implies x = 8 \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found by completing the square to transform it into the vertex form \( y - k = a(x - h)^2 \), where \( (h, k) \) is the vertex. Alternatively, the $x$-coordinate of the vertex is \( h = -\frac{b}{2a} \).

Step 2: Detailed Explanation:
Given equation: \( 2y = -3x^2 + 48x - 200 \).
Rewrite as: \( y = -\frac{3}{2}x^2 + 24x - 100 \).
Using the formula \( h = -\frac{b}{2a} \):
\( a = -\frac{3}{2} \), \( b = 24 \).
\[ h = -\frac{24}{2(-\frac{3}{2})} = -\frac{24}{-3} = 8 \]
Now, find the $y$-coordinate by substituting \( x = 8 \) into the original equation:
\[ 2y = -3(8)^2 + 48(8) - 200 \]
\[ 2y = -3(64) + 384 - 200 \]
\[ 2y = -192 + 384 - 200 \]
\[ 2y = 192 - 200 \]
\[ 2y = -8 \]
\[ y = -4 \]
Thus, the vertex is \( (8, -4) \).

Step 3: Final Answer:
The vertex of the parabola is $(8, -4)$.