Question: Exercise 7.08
1) Draw these graphs. Use a scale in radians on the x-axis.
10 y = cos(x + \pi)
11 y = sin(x + \pi/2)
12 y = sin(x - \pi/4)
13 y = cos(x - 3)
14 y = 1 - sin(x)
15 y = 2 sin(x) + 1
graph: multiple trigonometric function graphs listed in the center-right area; position_hint \in {center-right}
User: draw the trig graph on #10 only
1. **Problem:** Draw the graph of $$y = \cos(x + \pi)$$ using a scale in radians on the x-axis.
2. **Formula and properties:**
- The cosine function $$\cos(x)$$ has period $$2\pi$$ and amplitude 1.
- The function $$y = \cos(x + \pi)$$ is a horizontal shift of $$\cos(x)$$ by $$-\pi$$ units (shift left by $$\pi$$).
- Important identity: $$\cos(x + \pi) = -\cos(x)$$.
3. **Step-by-step graph transformation:**
- Start with the basic graph of $$y = \cos(x)$$.
- Apply the phase shift: replace $$x$$ by $$x + \pi$$.
- Using the identity, rewrite as $$y = -\cos(x)$$.
- This means the graph of $$\cos(x)$$ is reflected about the x-axis.
4. **Intermediate work:**
$$
y = \cos(x + \pi) \\
= -\cos(x)
$$
5. **Explanation:**
- The graph of $$y = \cos(x)$$ oscillates between 1 and -1 with zeros at $$x = \frac{\pi}{2} + k\pi$$.
- Reflecting it about the x-axis flips the peaks and troughs.
- So the graph of $$y = \cos(x + \pi)$$ has peaks at $$x = \pi + 2k\pi$$ where $$y = -1$$ and troughs at $$x = 2k\pi$$ where $$y = 1$$.
6. **Final answer:**
- The graph of $$y = \cos(x + \pi)$$ is the reflection of $$y = \cos(x)$$ about the x-axis.
- It has the same period $$2\pi$$ and amplitude 1.