Question

The graph of which trigonometric function(s) include the point left-parenthesis 0 comma 3 right-parenthesis? Select all that apply.
(2 points)
Responses

y = cos x + 3
y = cos x + 3

y = 3 cos x
y = 3 cos x

y = sin 2x + 3
y = sin 2 x + 3

y = tan x + 3
y = tan x + 3

f left parenthesis x right parenthesis equals sine left parenthesis 3 x plus Start Fraction pi over 2 End Fraction right parenthesis plus 1

Answer 1

To determine which trigonometric functions include the point \((0, 3)\), we need to evaluate the functions given at \(x = 0\) and see if the output is \(3\).

1. **\(y = \cos x + 3\)**:
- Substitute \(x = 0\):
\[
y = \cos(0) + 3 = 1 + 3 = 4
\]
- At \(x = 0\), \(y = 4\) (not \(3\)).

2. **\(y = 3 \cos x\)**:
- Substitute \(x = 0\):
\[
y = 3 \cos(0) = 3 \cdot 1 = 3
\]
- At \(x = 0\), \(y = 3\).

3. **\(y = \sin(2x) + 3\)**:
- Substitute \(x = 0\):
\[
y = \sin(2 \cdot 0) + 3 = \sin(0) + 3 = 0 + 3 = 3
\]
- At \(x = 0\), \(y = 3\).

4. **\(y = \tan x + 3\)**:
- Substitute \(x = 0\):
\[
y = \tan(0) + 3 = 0 + 3 = 3
\]
- At \(x = 0\), \(y = 3\).

5. **\(f(x) = \sin(3x + \frac{\pi}{2}) + 1\)**:
- Substitute \(x = 0\):
\[
f(0) = \sin\left(3 \cdot 0 + \frac{\pi}{2}\right) + 1 = \sin\left(\frac{\pi}{2}\right) + 1 = 1 + 1 = 2
\]
- At \(x = 0\), \(y = 2\) (not \(3\)).

**Summary of results**:
- \(y = 3 \cos x\) gives \(3\) at \((0, 3)\).
- \(y = \sin(2x) + 3\) gives \(3\) at \((0, 3)\).
- \(y = \tan x + 3\) gives \(3\) at \((0, 3)\).

Therefore, the functions that include the point \((0, 3)\) are:
- \(y = 3 \cos x\)
- \(y = \sin(2x) + 3\)
- \(y = \tan x + 3\)