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

Question

(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 $x=0$: $$f(0)=\sin (3\cdot 0+\frac{\pi }{2})+1=\sin \left(\frac{\pi }{2}\right)+1=1+1=2$$
   • 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:

• $y=3\cos x$
• $y=\sin (2x)+3$
• $y=\tan x+3$