$$ y=2 \tan \left(\frac{1}{2} x+\frac{\pi}{4}\right)-1 $$ Drag the movable black point to shift the function, the red points to set the vertical asymptotes, and the blue point at the correct set of coordinates. You may click on a point to verify its coordinates. Note that only one period of the function is shown.

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### Step 1: Write the Function in Standard Form

We start with the function

$$
y = 2 \tan\left(\frac{1}{2}x+\frac{\pi}{4}\right)-1.
$$

Notice that the argument of the tangent can be factored as

$$
\frac{1}{2}x+\frac{\pi}{4}=\frac{1}{2}\left(x+\frac{\pi}{2}\right),
$$

which shows a horizontal stretch by a factor of 2 and a horizontal shift to the left by $$\frac{\pi}{2}$$. The $$-1$$ at the end represents a vertical shift downward by 1 unit.

---

### Step 2: Find the Vertical Asymptotes

For the tangent function, vertical asymptotes occur when the argument is

$$
\frac{\pi}{2}+k\pi,\quad k\in\mathbb{Z}.
$$

Set the inner expression equal to this value:

$$
\frac{1}{2}x+\frac{\pi}{4}=\frac{\pi}{2}+k\pi.
$$

Subtract $$\frac{\pi}{4}$$ from both sides:

$$
\frac{1}{2}x=\frac{\pi}{2}-\frac{\pi}{4}+k\pi=\frac{\pi}{4}+k\pi.
$$

Multiply by 2:

$$
x=\frac{\pi}{2}+2k\pi.
$$

For one period of the function, choose two consecutive values of $$k$$. For example, using $$k=-1$$ and $$k=0$$ gives the vertical asymptotes:

$$
x=\frac{\pi}{2}+2(-1)\pi=\frac{\pi}{2}-2\pi=-\frac{3\pi}{2},
$$
$$
x=\frac{\pi}{2}+2(0)\pi=\frac{\pi}{2}.
$$

Thus, the red points (vertical asymptotes) should be placed at $$ x=-\frac{3\pi}{2} $$ and $$ x=\frac{\pi}{2} $$.

---

### Step 3: Locate a Key Point on the Graph

A good point to find is when $$x=0$$, as it lies between the two vertical asymptotes. Substitute $$x=0$$ into the expression:

$$
y=2 \tan\left(\frac{1}{2}(0)+\frac{\pi}{4}\right)-1=2 \tan\left(\frac{\pi}{4}\right)-1.
$$

Since

$$
\tan\left(\frac{\pi}{4}\right)=1,
$$

we obtain

$$
y=2\cdot 1-1=2-1=1.
$$

Thus, the blue point should have the coordinates $$(0,1)$$.

---

### Step 4: Adjust the Movable Points

- **Black Point:** This point is used to shift the function into position as needed. In this case, the function already incorporates a horizontal shift (left by $$\frac{\pi}{2}$$) and a vertical shift (down by 1), so dragging the black point ensures the correct overall placement of the curve.
- **Red Points:** Place them at $$ x=-\frac{3\pi}{2} $$ and $$ x=\frac{\pi}{2} $$ to mark the vertical asymptotes.
- **Blue Point:** Set it at $$(0,1)$$ because it lies on the graph of the function.

These adjustments ensure that only one period of the function is visible and correctly positioned.

To graph the function $$ y = 2 \tan\left(\frac{1}{2}x + \frac{\pi}{4}\right) - 1 $$: 1. **Vertical Asymptotes**: Place red points at $$ x = -\frac{3\pi}{2} $$ and $$ x = \frac{\pi}{2} $$. 2. **Key Point**: Set the blue point at $$(0, 1)$$. 3. **Shift**: Use the black point to adjust the graph so that it correctly shows one period with the vertical asymptotes and the key point in place. This setup ensures the function is accurately represented within one period.

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