The graph of the function f(x) = A \sin(Bx + C) + D has a minimum point at \left(\frac{\pi… — ACT

The graph of the function $f(x) = A \sin(Bx + C) + D$ has a minimum point at $\left(\frac{\pi}{4}, -2\right)$ and its consecutive maximum point at $\left(\frac{5\pi}{4}, 8\right)$ , where $A > 0$ and $B > 0$ . What is the value of $f\left(\frac{7\pi}{4}\right)$ ?

A
-2
B
0
3Answer
D
5
E
8

Answer

The midline of a sinusoidal curve is the average of its maximum and minimum values, which is

\frac{8 + (-2)}{2} = 3

. The horizontal distance between consecutive minimum and maximum points is half of a period:

\frac{5\pi}{4} - \frac{\pi}{4} = \pi

. Thus, the full period is

2\pi

, and the next minimum occurs at

\frac{5\pi}{4} + \pi = \frac{9\pi}{4}

. The value

x = \frac{7\pi}{4}

lies halfway between the maximum at

\frac{5\pi}{4}

and the minimum at

\frac{9\pi}{4}

. Midway between a maximum and a minimum, the function crosses its midline, so

f\left(\frac{7\pi}{4}\right) = 3

Step-by-Step Solution

Determine the midline (vertical shift

D

) of the trigonometric function.

The midline is

D = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{8 + (-2)}{2} = 3

The midline lies exactly halfway between the vertical peak and trough.

Find the horizontal distance for a half-period and determine the full period.

Half-period =

\frac{5\pi}{4} - \frac{\pi}{4} = \pi

, so the full period

T = 2\pi

The horizontal distance between consecutive minimum and maximum points equals half of one full period.

Identify the behavior of the graph at

x = \frac{7\pi}{4}

Since

x = \frac{7\pi}{4}

is halfway between the maximum at

x = \frac{5\pi}{4}

and the next minimum at

x = \frac{9\pi}{4}

, the function value equals the midline height

D = 3

A sinusoidal wave crosses its midline exactly midway between a peak and a trough.

Key Concept

Key features of transformed trigonometric functions (amplitude, midline, period, and symmetry)

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic