The roller coaster is represented by the function f (x) = −16x2 + 128x + 50 where f (x) is the

Question

height of the roller coaster in feet above the ground after x seconds since the roller coaster
started up the first hill.
Which ordered pair represents the time the coaster reaches the
maximum height on the first hill and the maximum height?
A. (4, 306)
B. (4, 50)
C. (8, 50)
D. (8, 306)

Answer 1

To find the maximum height of the roller coaster represented by the function \( f(x) = -16x^2 + 128x + 50 \), we need to find the vertex of the parabola. The vertex formula for a quadratic function \( ax^2 + bx + c \) is given by:

\[ x = -\frac{b}{2a} \]

In this case, the coefficients are:

• \( a = -16 \)
• \( b = 128 \)
• \( c = 50 \)

Substituting the values of \( a \) and \( b \) into the vertex formula:

\[ x = -\frac{128}{2 \times -16} = -\frac{128}{-32} = 4 \]

Now, we substitute \( x = 4 \) back into the function \( f(x) \) to find the maximum height:

\[ f(4) = -16(4^2) + 128(4) + 50 \] \[ = -16(16) + 128(4) + 50 \] \[ = -256 + 512 + 50 \] \[ = 306 \]

Thus, the ordered pair representing the time the coaster reaches the maximum height and the maximum height is:

\[ (4, 306) \]

Therefore, the correct answer is:

A. (4, 306)