Question: Position the five points so they lie on the graph of the same sine function and so not more than two of them have the same y-value. Shifts along the x-axis are not allowed.
x
y
$-2\pi$, $-\frac{3\pi}{2}$, $-\pi$, $-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
Five black filled circles are stacked vertically at $x = \frac{\pi}{2}$ with y-values 1, 2, 3, 4, 5; position_hint=center
1. **State the problem:** We need to position five points on the graph of the same sine function $y = A\sin(x)$ without shifting along the x-axis, such that no more than two points share the same $y$-value.
2. **Recall the sine function properties:** The sine function $\sin(x)$ has a period of $2\pi$ and ranges between $-1$ and $1$. To fit points with different $y$-values, we can scale the amplitude $A$.
3. **Constraints:**
- No horizontal shifts allowed, so points must be at given $x$ values.
- At most two points can have the same $y$-value.
4. **Choose five distinct $x$ values from the given set:**
Let's pick $x = -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, \frac{\pi}{2}, \pi$.
5. **Calculate sine values at these points:**
$$\sin\left(-\frac{3\pi}{2}\right) = 1$$
$$\sin(-\pi) = 0$$
$$\sin\left(-\frac{\pi}{2}\right) = -1$$
$$\sin\left(\frac{\pi}{2}\right) = 1$$
$$\sin(\pi) = 0$$
6. **Note repeated $y$-values:**
- $\sin(-\frac{3\pi}{2}) = \sin(\frac{\pi}{2}) = 1$
- $\sin(-\pi) = \sin(\pi) = 0$
- $\sin(-\frac{\pi}{2}) = -1$
7. **Scale amplitude $A$ to get desired $y$-values:**
We want five points with $y$-values 1, 2, 3, 4, 5 but sine ranges only $[-1,1]$. So set $A=5$ to scale sine values:
$$y = 5\sin(x)$$
8. **Calculate scaled $y$-values:**
$$y(-\frac{3\pi}{2}) = 5 \times 1 = 5$$
$$y(-\pi) = 5 \times 0 = 0$$
$$y(-\frac{\pi}{2}) = 5 \times (-1) = -5$$
$$y(\frac{\pi}{2}) = 5 \times 1 = 5$$
$$y(\pi) = 5 \times 0 = 0$$
9. **Adjust points to avoid more than two with same $y$-value:**
Currently, $y=5$ and $y=0$ each appear twice, $y=-5$ once.
10. **Select points with distinct $y$-values:**
Choose $x = -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, \frac{\pi}{2}, \pi$ with $y$-values $5, 0, -5, 5, 0$ respectively.
11. **Final positioning:**
- At $x = -\frac{3\pi}{2}$, $y=5$
- At $x = -\pi$, $y=0$
- At $x = -\frac{\pi}{2}$, $y=-5$
- At $x = \frac{\pi}{2}$, $y=5$
- At $x = \pi$, $y=0$
This satisfies the condition that no more than two points share the same $y$-value and all points lie on $y=5\sin(x)$ without horizontal shifts.
**Answer:** The five points lie on the graph of $y=5\sin(x)$ at $x = -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, \frac{\pi}{2}, \pi$ with corresponding $y$-values $5, 0, -5, 5, 0$.