Question
Gerard made a table for the function LaTeX: y=-2cos(4x+\frac{\pi}{2})-7
x
y
0
-9
4
-7
38
-5
2
-7
58
-9
What was his error?
x
y
0
-9
4
-7
38
-5
2
-7
58
-9
What was his error?
Answers
GPT-4o mini
To determine if there is an error in the values presented for the function \( y = -2 \cos(4x + \frac{\pi}{2}) - 7 \), we need to evaluate the function at the specified \( x \) values provided and compare the results with the corresponding \( y \) values in the table.
1. **Understanding the cosine function**:
The cosine function \( \cos(z) \) has a periodicity of \( 2\pi \) and outputs values in the range \([-1, 1]\). The transformation \( -2 \cos(z) \) will output values in the range \([-2, 2]\), and then shifting by \(-7\) will change this to \([-9, -5]\).
2. **Evaluating the function at the specified \( x \) values**:
- **For \( x = 0 \)**:
\[
y = -2 \cos(4(0) + \frac{\pi}{2}) - 7 = -2 \cos\left(\frac{\pi}{2}\right) - 7 = -2(0) - 7 = -7
\]
- **For \( x = 4 \)**:
\[
y = -2 \cos(4(4) + \frac{\pi}{2}) = -2 \cos(16 + \frac{\pi}{2}) = -2 \cos(16 + \frac{\pi}{2}) = -2(-\sin(16)) - 7
\]
Calculating \( \cos(16 + \frac{\pi}{2}) = -\sin(16) \) requires knowing the sine of 16 radians. (We'll note it isn't required to finalize the answer, just evaluating might be complex).
- **For \( x = 38 \)**:
\[
y = -2 \cos(4(38) + \frac{\pi}{2}) = -2 \cos(152 + \frac{\pi}{2}) = -2(-\sin(152)) - 7
\]
- **For \( x = 2 \)**:
\[
y = -2 \cos(4(2) + \frac{\pi}{2}) = -2 \cos(8 + \frac{\pi}{2}) = -2(-\sin(8)) - 7
\]
3. **Comparing the values from the table with the calculated values**:
- For \( x = 0 \): **Table says -9**, but calculated gives **-7** → Error here.
- For \( x = 4 \): **Table says -7**, will need sine value for verification.
- For \( x = 38 \): **Table says -5**, again will need sine value.
- For \( x = 2 \): **Table says -7**, also needs sine value.
From preliminary checks:
- **The value for \( x=0 \)** was identified as an error in his table where \( y \) should be **-7**, not **-9**.
**Conclusion**: Gerard's primary error is in the calculated value for \( y \) when \( x = 0\). The correct value is -7, not -9. Other evaluations should be checked for correctness, but the significant error is noted initially here.
1. **Understanding the cosine function**:
The cosine function \( \cos(z) \) has a periodicity of \( 2\pi \) and outputs values in the range \([-1, 1]\). The transformation \( -2 \cos(z) \) will output values in the range \([-2, 2]\), and then shifting by \(-7\) will change this to \([-9, -5]\).
2. **Evaluating the function at the specified \( x \) values**:
- **For \( x = 0 \)**:
\[
y = -2 \cos(4(0) + \frac{\pi}{2}) - 7 = -2 \cos\left(\frac{\pi}{2}\right) - 7 = -2(0) - 7 = -7
\]
- **For \( x = 4 \)**:
\[
y = -2 \cos(4(4) + \frac{\pi}{2}) = -2 \cos(16 + \frac{\pi}{2}) = -2 \cos(16 + \frac{\pi}{2}) = -2(-\sin(16)) - 7
\]
Calculating \( \cos(16 + \frac{\pi}{2}) = -\sin(16) \) requires knowing the sine of 16 radians. (We'll note it isn't required to finalize the answer, just evaluating might be complex).
- **For \( x = 38 \)**:
\[
y = -2 \cos(4(38) + \frac{\pi}{2}) = -2 \cos(152 + \frac{\pi}{2}) = -2(-\sin(152)) - 7
\]
- **For \( x = 2 \)**:
\[
y = -2 \cos(4(2) + \frac{\pi}{2}) = -2 \cos(8 + \frac{\pi}{2}) = -2(-\sin(8)) - 7
\]
3. **Comparing the values from the table with the calculated values**:
- For \( x = 0 \): **Table says -9**, but calculated gives **-7** → Error here.
- For \( x = 4 \): **Table says -7**, will need sine value for verification.
- For \( x = 38 \): **Table says -5**, again will need sine value.
- For \( x = 2 \): **Table says -7**, also needs sine value.
From preliminary checks:
- **The value for \( x=0 \)** was identified as an error in his table where \( y \) should be **-7**, not **-9**.
**Conclusion**: Gerard's primary error is in the calculated value for \( y \) when \( x = 0\). The correct value is -7, not -9. Other evaluations should be checked for correctness, but the significant error is noted initially here.