Question

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)

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.