Using the function h(x)=sin(−4x) , find the y-values in the second row of the table by substituting each x-value into the function. Which of the following options shows the graph of h(x) ? x −π/2 −π/4 0 π/2 h(x)

Question

what does the graph ook like

Answer 1

To find the \( y \)-values in the table for the function \( h(x) = \sin(-4x) \), we will substitute each provided \( x \)-value into the function.

Given \( x \)-values: \[ -\frac{\pi}{2}, -\frac{\pi}{4}, 0, \frac{\pi}{2} \]

Let's calculate \( h(x) \) for each \( x \).

1. For \( x = -\frac{\pi}{2} \): \[ h\left(-\frac{\pi}{2}\right) = \sin\left(-4 \times -\frac{\pi}{2}\right) = \sin(2\pi) = 0 \]

2. For \( x = -\frac{\pi}{4} \): \[ h\left(-\frac{\pi}{4}\right) = \sin\left(-4 \times -\frac{\pi}{4}\right) = \sin(\pi) = 0 \]

3. For \( x = 0 \): \[ h(0) = \sin(-4 \times 0) = \sin(0) = 0 \]

4. For \( x = \frac{\pi}{2} \): \[ h\left(\frac{\pi}{2}\right) = \sin\left(-4 \times \frac{\pi}{2}\right) = \sin(-2\pi) = 0 \]

Now we have the \( y \)-values for each of the \( x \)-values:

\[ \begin{array}{c|c} x & h(x) \ \hline -\frac{\pi}{2} & 0 \ -\frac{\pi}{4} & 0 \ 0 & 0 \ \frac{\pi}{2} & 0 \ \end{array} \]

Summary of Values

• \( h\left(-\frac{\pi}{2}\right) = 0 \)
• \( h\left(-\frac{\pi}{4}\right) = 0 \)
• \( h(0) = 0 \)
• \( h\left(\frac{\pi}{2}\right) = 0 \)

Graph of \( h(x) = \sin(-4x) \)

The function \( h(x) = \sin(-4x) \) will have a period of \( \frac{\pi}{2} \) because the coefficient \( -4 \) in front of \( x \) compresses the sine wave by a factor of 4.

• The sine function oscillates between -1 and 1.
• As such, in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), the graph will have points at \( (0, 0) \), and since it is zero at all the points we've calculated, it implies the function will cross the x-axis at these intervals.

Looking at the Graph

The graph of \( h(x) \) will be a sine wave oscillating about the x-axis with the following characteristics:

• It will have peaks at \( y = 1 \) and troughs at \( y = -1 \).
• The function will cross the x-axis at the calculated points, specifically \( x = -\frac{\pi}{2}, -\frac{\pi}{4}, 0, \frac{\pi}{2} \).

In conclusion, you would find a graph that resembles a compressed sine wave with periodic crossings at the x-axis at those specified values, possibly repeating in both the positive and negative directions.