Answer :
We are given the following properties for a function and need to identify which of the four provided functions meets these criteria.
1. The domain is the set of all real numbers.
2. One [tex]\( x \)[/tex]-intercept is [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex].
3. The maximum value is 3.
4. The [tex]\( y \)[/tex]-intercept is [tex]\((0, -3)\)[/tex].
Let's analyze each function step-by-step according to the given properties.
### Option 1: [tex]\( y = -3 \sin(x) \)[/tex]
1. Domain: The sine function is defined for all real numbers, so this criterion is met.
2. [tex]\(x\)[/tex]-intercept: When [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[ y = -3 \sin\left(\frac{\pi}{2}\right) = -3 \cdot 1 = -3 \][/tex]
This does not give us [tex]\((\frac{\pi}{2}, 0)\)[/tex].
3. Maximum value: The maximum value of [tex]\(-3 \sin(x)\)[/tex] occurs when [tex]\(\sin(x) = -1\)[/tex]:
[tex]\[ \max = -3 \times -1 = 3 \][/tex]
This criterion is met.
4. [tex]\(y\)[/tex]-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3 \sin(0) = -3 \cdot 0 = 0 \][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\((0, 0)\)[/tex], not [tex]\((0, -3)\)[/tex].
Therefore, [tex]\( y = -3 \sin(x) \)[/tex] does not meet all the given properties.
### Option 2: [tex]\( y = -3 \cos(x) \)[/tex]
1. Domain: The cosine function is defined for all real numbers, so this criterion is met.
2. [tex]\(x\)[/tex]-intercept: When [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[ y = -3 \cos\left(\frac{\pi}{2}\right) = -3 \cdot 0 = 0 \][/tex]
This gives the correct [tex]\( x \)[/tex]-intercept [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex].
3. Maximum value: The maximum value of [tex]\(-3 \cos(x)\)[/tex] occurs when [tex]\(\cos(x) = -1\)[/tex]:
[tex]\[ \max = -3 \times -1 = 3 \][/tex]
This criterion is met.
4. [tex]\(y\)[/tex]-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3 \cos(0) = -3 \cdot 1 = -3 \][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\((0, -3)\)[/tex].
Thus, [tex]\( y = -3 \cos(x) \)[/tex] meets all the given properties.
### Option 3: [tex]\( y = 3 \sin(x) \)[/tex]
1. Domain: The sine function is defined for all real numbers, so this criterion is met.
2. [tex]\(x\)[/tex]-intercept: When [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[ y = 3 \sin\left(\frac{\pi}{2}\right) = 3 \times 1 = 3 \][/tex]
This does not give us [tex]\((\frac{\pi}{2}, 0)\)[/tex].
3. Maximum value: The maximum value of [tex]\(3 \sin(x)\)[/tex] occurs when [tex]\(\sin(x) = 1\)[/tex]:
[tex]\[ \max = 3 \times 1 = 3 \][/tex]
This criterion is met.
4. [tex]\(y\)[/tex]-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3 \sin(0) = 3 \times 0 = 0 \][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\((0, 0)\)[/tex], not [tex]\((0, -3)\)[/tex].
Therefore, [tex]\( y = 3 \sin(x) \)[/tex] does not meet all the given properties.
### Option 4: [tex]\( y = 3 \cos(x) \)[/tex]
1. Domain: The cosine function is defined for all real numbers, so this criterion is met.
2. [tex]\(x\)[/tex]-intercept: When [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[ y = 3 \cos\left(\frac{\pi}{2}\right) = 3 \cdot 0 = 0 \][/tex]
This gives the correct [tex]\( x \)[/tex]-intercept [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex].
3. Maximum value: The maximum value of [tex]\(3 \cos(x)\)[/tex] occurs when [tex]\(\cos(x) = 1\)[/tex]:
[tex]\[ \max = 3 \times 1 = 3 \][/tex]
This criterion is met.
4. [tex]\(y\)[/tex]-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3 \cos(0) = 3 \cdot 1 = 3 \][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\((0, 3)\)[/tex], not [tex]\((0, -3)\)[/tex].
Therefore, [tex]\( y = 3 \cos(x) \)[/tex] does not meet all the given properties.
### Conclusion
None of the functions meets all the specified properties, hence none of the given functions match the criteria.
1. The domain is the set of all real numbers.
2. One [tex]\( x \)[/tex]-intercept is [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex].
3. The maximum value is 3.
4. The [tex]\( y \)[/tex]-intercept is [tex]\((0, -3)\)[/tex].
Let's analyze each function step-by-step according to the given properties.
### Option 1: [tex]\( y = -3 \sin(x) \)[/tex]
1. Domain: The sine function is defined for all real numbers, so this criterion is met.
2. [tex]\(x\)[/tex]-intercept: When [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[ y = -3 \sin\left(\frac{\pi}{2}\right) = -3 \cdot 1 = -3 \][/tex]
This does not give us [tex]\((\frac{\pi}{2}, 0)\)[/tex].
3. Maximum value: The maximum value of [tex]\(-3 \sin(x)\)[/tex] occurs when [tex]\(\sin(x) = -1\)[/tex]:
[tex]\[ \max = -3 \times -1 = 3 \][/tex]
This criterion is met.
4. [tex]\(y\)[/tex]-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3 \sin(0) = -3 \cdot 0 = 0 \][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\((0, 0)\)[/tex], not [tex]\((0, -3)\)[/tex].
Therefore, [tex]\( y = -3 \sin(x) \)[/tex] does not meet all the given properties.
### Option 2: [tex]\( y = -3 \cos(x) \)[/tex]
1. Domain: The cosine function is defined for all real numbers, so this criterion is met.
2. [tex]\(x\)[/tex]-intercept: When [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[ y = -3 \cos\left(\frac{\pi}{2}\right) = -3 \cdot 0 = 0 \][/tex]
This gives the correct [tex]\( x \)[/tex]-intercept [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex].
3. Maximum value: The maximum value of [tex]\(-3 \cos(x)\)[/tex] occurs when [tex]\(\cos(x) = -1\)[/tex]:
[tex]\[ \max = -3 \times -1 = 3 \][/tex]
This criterion is met.
4. [tex]\(y\)[/tex]-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3 \cos(0) = -3 \cdot 1 = -3 \][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\((0, -3)\)[/tex].
Thus, [tex]\( y = -3 \cos(x) \)[/tex] meets all the given properties.
### Option 3: [tex]\( y = 3 \sin(x) \)[/tex]
1. Domain: The sine function is defined for all real numbers, so this criterion is met.
2. [tex]\(x\)[/tex]-intercept: When [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[ y = 3 \sin\left(\frac{\pi}{2}\right) = 3 \times 1 = 3 \][/tex]
This does not give us [tex]\((\frac{\pi}{2}, 0)\)[/tex].
3. Maximum value: The maximum value of [tex]\(3 \sin(x)\)[/tex] occurs when [tex]\(\sin(x) = 1\)[/tex]:
[tex]\[ \max = 3 \times 1 = 3 \][/tex]
This criterion is met.
4. [tex]\(y\)[/tex]-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3 \sin(0) = 3 \times 0 = 0 \][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\((0, 0)\)[/tex], not [tex]\((0, -3)\)[/tex].
Therefore, [tex]\( y = 3 \sin(x) \)[/tex] does not meet all the given properties.
### Option 4: [tex]\( y = 3 \cos(x) \)[/tex]
1. Domain: The cosine function is defined for all real numbers, so this criterion is met.
2. [tex]\(x\)[/tex]-intercept: When [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[ y = 3 \cos\left(\frac{\pi}{2}\right) = 3 \cdot 0 = 0 \][/tex]
This gives the correct [tex]\( x \)[/tex]-intercept [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex].
3. Maximum value: The maximum value of [tex]\(3 \cos(x)\)[/tex] occurs when [tex]\(\cos(x) = 1\)[/tex]:
[tex]\[ \max = 3 \times 1 = 3 \][/tex]
This criterion is met.
4. [tex]\(y\)[/tex]-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3 \cos(0) = 3 \cdot 1 = 3 \][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\((0, 3)\)[/tex], not [tex]\((0, -3)\)[/tex].
Therefore, [tex]\( y = 3 \cos(x) \)[/tex] does not meet all the given properties.
### Conclusion
None of the functions meets all the specified properties, hence none of the given functions match the criteria.
