Question: Question 27
0/1 pt 2 99 Details
4
3
2
1
-1
-2
-3
-4
-3/2
-3/4
3/4
3/2
9/4
3
15/4
x
Above is a graph of a shifted sinusoidal function. State the function's formulas if the function was originally a regular (positive) sine and cosine function. Give EXACT answers -- no decimals!
As a shifted cosine function: y =
As a shifted sine function: y =
Question Help: Message instructor Post to forum
Graph: a blue sinusoidal curve with midline y = 0, amplitude 1, and period 3, crossing the x-axis at x = -3/2, -3/4, 3/4, 3/2, 9/4, 3, and 15/4; it has a maximum near ( -3/4, 1 ) and again near ( 9/4, 1 ), and a minimum near ( 3/4, -1 ) and again near ( 3, -1 ); position_hint = centerwhats answer
1. **State the problem:**
We have a sinusoidal function graphed with amplitude 1, midline $y=0$, and period 3. We want to find the exact formulas for this function as a shifted cosine and a shifted sine function.
2. **Recall the general forms:**
- Cosine form: $$y = A \cos\left(B(x - C)\right) + D$$
- Sine form: $$y = A \sin\left(B(x - C)\right) + D$$
Where:
- $A$ is amplitude
- $B = \frac{2\pi}{\text{period}}$
- $C$ is horizontal shift
- $D$ is vertical shift (midline)
3. **Identify given values:**
- Amplitude $A = 1$
- Midline $D = 0$
- Period $= 3$ so $B = \frac{2\pi}{3}$
4. **Find horizontal shift $C$ for cosine:**
Cosine normally has a maximum at $x=0$. The graph's maximum is at $x = -\frac{3}{4}$.
So, for cosine:
$$x - C = 0 \implies C = -\frac{3}{4}$$
Thus, the cosine function is:
$$y = \cos\left(\frac{2\pi}{3}\left(x + \frac{3}{4}\right)\right)$$
5. **Find horizontal shift $C$ for sine:**
Sine normally has a zero crossing at $x=0$ going upward. The graph crosses zero at $x = -\frac{3}{2}$ and $x = -\frac{3}{4}$, but the zero crossing with positive slope is at $x = -\frac{3}{2}$.
Sine zero at $x=0$ means:
$$x - C = 0 \implies C = -\frac{3}{2}$$
So the sine function is:
$$y = \sin\left(\frac{2\pi}{3}\left(x + \frac{3}{2}\right)\right)$$
6. **Final answers:**
- As a shifted cosine function:
$$y = \cos\left(\frac{2\pi}{3}\left(x + \frac{3}{4}\right)\right)$$
- As a shifted sine function:
$$y = \sin\left(\frac{2\pi}{3}\left(x + \frac{3}{2}\right)\right)$$