Question
Using h(x)=−2sinx , find the y-values in the second row of the table by substituting each x-value into the function. Which of the following is the graph of h(x) ?
This table displays given x-values, where the corresponding h(x) values need to be calculated to complete the table.
x −3π2 −π2 π2 3π2
h(x)
(1 point)
Responses
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
Image with alt text: A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
Image with alt text: A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
Image with alt text: A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
This table displays given x-values, where the corresponding h(x) values need to be calculated to complete the table.
x −3π2 −π2 π2 3π2
h(x)
(1 point)
Responses
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
Image with alt text: A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
Image with alt text: A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
Image with alt text: A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
A waveform is graphed on a coordinate plane. The x-axis ranges from negative 2 pi to 2 pi in increments of Start Fraction pi over 2 End Fraction. The y-axis ranges from negative 2 to 2 in increments of 1.
Answers
GPT-4o mini
To find the y-values of the function \( h(x) = -2\sin(x) \) for the given x-values, we will substitute each x-value into the function.
### Given x-values:
- \( x = -\frac{3\pi}{2} \)
- \( x = -\frac{\pi}{2} \)
- \( x = \frac{\pi}{2} \)
- \( x = \frac{3\pi}{2} \)
### Calculations:
1. **For \( x = -\frac{3\pi}{2} \)**:
\[
h\left(-\frac{3\pi}{2}\right) = -2\sin\left(-\frac{3\pi}{2}\right)
\]
\[
\sin\left(-\frac{3\pi}{2}\right) = -1 \quad \text{(since } \sin\left(-\frac{3\pi}{2}\right) \text{ is at the angle } \frac{3\pi}{2} \text{ which is } -1 \text{)}
\]
\[
h\left(-\frac{3\pi}{2}\right) = -2(-1) = 2
\]
2. **For \( x = -\frac{\pi}{2} \)**:
\[
h\left(-\frac{\pi}{2}\right) = -2\sin\left(-\frac{\pi}{2}\right)
\]
\[
\sin\left(-\frac{\pi}{2}\right) = -1
\]
\[
h\left(-\frac{\pi}{2}\right) = -2(-1) = 2
\]
3. **For \( x = \frac{\pi}{2} \)**:
\[
h\left(\frac{\pi}{2}\right) = -2\sin\left(\frac{\pi}{2}\right)
\]
\[
\sin\left(\frac{\pi}{2}\right) = 1
\]
\[
h\left(\frac{\pi}{2}\right) = -2(1) = -2
\]
4. **For \( x = \frac{3\pi}{2} \)**:
\[
h\left(\frac{3\pi}{2}\right) = -2\sin\left(\frac{3\pi}{2}\right)
\]
\[
\sin\left(\frac{3\pi}{2}\right) = -1
\]
\[
h\left(\frac{3\pi}{2}\right) = -2(-1) = 2
\]
### Summary of Results:
- \( h\left(-\frac{3\pi}{2}\right) = 2 \)
- \( h\left(-\frac{\pi}{2}\right) = 2 \)
- \( h\left(\frac{\pi}{2}\right) = -2 \)
- \( h\left(\frac{3\pi}{2}\right) = 2 \)
### Completing the Table:
Now, we can fill in the table with the calculated \( h(x) \) values:
| x | \( -\frac{3\pi}{2} \) | \( -\frac{\pi}{2} \) | \( \frac{\pi}{2} \) | \( \frac{3\pi}{2} \) |
|--------------|-----------------------|---------------------|--------------------|---------------------|
| h(x) | 2 | 2 | -2 | 2 |
### Determining the Graph:
The function \( h(x) = -2\sin(x) \) produces a sinusoidal wave that:
- Has a maximum value of 2 and a minimum value of -2.
- Is inverted, meaning the peaks are at the troughs of a regular sine wave and vice versa.
The graph you should select is the one that shows this waveform behavior with maximum and minimum values occurring at the specified points. It should exhibit the characteristics described above within the given range of the axes.
If you provided the visual images of the graphs, then you could select the one that represents these characteristics most accurately.
### Given x-values:
- \( x = -\frac{3\pi}{2} \)
- \( x = -\frac{\pi}{2} \)
- \( x = \frac{\pi}{2} \)
- \( x = \frac{3\pi}{2} \)
### Calculations:
1. **For \( x = -\frac{3\pi}{2} \)**:
\[
h\left(-\frac{3\pi}{2}\right) = -2\sin\left(-\frac{3\pi}{2}\right)
\]
\[
\sin\left(-\frac{3\pi}{2}\right) = -1 \quad \text{(since } \sin\left(-\frac{3\pi}{2}\right) \text{ is at the angle } \frac{3\pi}{2} \text{ which is } -1 \text{)}
\]
\[
h\left(-\frac{3\pi}{2}\right) = -2(-1) = 2
\]
2. **For \( x = -\frac{\pi}{2} \)**:
\[
h\left(-\frac{\pi}{2}\right) = -2\sin\left(-\frac{\pi}{2}\right)
\]
\[
\sin\left(-\frac{\pi}{2}\right) = -1
\]
\[
h\left(-\frac{\pi}{2}\right) = -2(-1) = 2
\]
3. **For \( x = \frac{\pi}{2} \)**:
\[
h\left(\frac{\pi}{2}\right) = -2\sin\left(\frac{\pi}{2}\right)
\]
\[
\sin\left(\frac{\pi}{2}\right) = 1
\]
\[
h\left(\frac{\pi}{2}\right) = -2(1) = -2
\]
4. **For \( x = \frac{3\pi}{2} \)**:
\[
h\left(\frac{3\pi}{2}\right) = -2\sin\left(\frac{3\pi}{2}\right)
\]
\[
\sin\left(\frac{3\pi}{2}\right) = -1
\]
\[
h\left(\frac{3\pi}{2}\right) = -2(-1) = 2
\]
### Summary of Results:
- \( h\left(-\frac{3\pi}{2}\right) = 2 \)
- \( h\left(-\frac{\pi}{2}\right) = 2 \)
- \( h\left(\frac{\pi}{2}\right) = -2 \)
- \( h\left(\frac{3\pi}{2}\right) = 2 \)
### Completing the Table:
Now, we can fill in the table with the calculated \( h(x) \) values:
| x | \( -\frac{3\pi}{2} \) | \( -\frac{\pi}{2} \) | \( \frac{\pi}{2} \) | \( \frac{3\pi}{2} \) |
|--------------|-----------------------|---------------------|--------------------|---------------------|
| h(x) | 2 | 2 | -2 | 2 |
### Determining the Graph:
The function \( h(x) = -2\sin(x) \) produces a sinusoidal wave that:
- Has a maximum value of 2 and a minimum value of -2.
- Is inverted, meaning the peaks are at the troughs of a regular sine wave and vice versa.
The graph you should select is the one that shows this waveform behavior with maximum and minimum values occurring at the specified points. It should exhibit the characteristics described above within the given range of the axes.
If you provided the visual images of the graphs, then you could select the one that represents these characteristics most accurately.