Graph cosine functions worksheet with six problems, each requiring students to graph a given cosine function on a provided coordinate grid.

Here are the step-by-step solutions for graphing each cosine function. To graph these, we look at four main features:
1. Midline (Vertical Shift): The number added or subtracted at the end moves the center line of the wave up or down.
2. Amplitude: The number multiplying the cosine tells us how far the graph goes up and down from the midline. If it's negative, the graph flips upside down.
3. Period: The number inside next to $x$ changes how stretched or squished the wave is horizontally.
4. Phase Shift: The number added or subtracted inside the parenthesis moves the graph left or right.

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

* Midline: $y = -2$ (Shift down 2).
* Amplitude: $1$ (Standard height). Max is $-1$, Min is $-3$.
* Phase Shift: Left $\frac{\pi}{4}$. A standard cosine starts at a maximum. So, this graph starts its maximum at $x = -\frac{\pi}{4}$.
* Key Points to Plot:
* Start (Max): $(-\frac{\pi}{4}, -1)$
* Midline (going down): $(\frac{\pi}{4}, -2)$
* Minimum: $(\frac{3\pi}{4}, -3)$
* Midline (going up): $(\frac{5\pi}{4}, -2)$
* End (Max): $(\frac{7\pi}{4}, -1)$

2. $y = \cos(x + \pi) + 5$

* Midline: $y = 5$ (Shift up 5).
* Amplitude: $1$. Max is $6$, Min is $4$.
* Phase Shift: Left $\pi$. Standard max is at $0$, so new max is at $-\pi$.
* *Note:* $\cos(x + \pi)$ is actually the same as $-\cos(x)$, which means it's an upside-down cosine centered at the origin.
* Key Points to Plot:
* Max: $(-\pi, 6)$ and $(\pi, 6)$
* Min: $(0, 4)$
* Midline: $(-\frac{\pi}{2}, 5)$ and $(\frac{\pi}{2}, 5)$

3. $y = \cos(x - \frac{\pi}{2}) + 3$

* Midline: $y = 3$ (Shift up 3).
* Amplitude: $1$. Max is $4$, Min is $2$.
* Phase Shift: Right $\frac{\pi}{2}$. Standard max is at $0$, so new max is at $\frac{\pi}{2}$.
* *Note:* This is identical to the sine function ($y = \sin(x) + 3$). It starts at the midline going up at $x=0$.
* Key Points to Plot:
* Midline (start): $(0, 3)$
* Max: $(\frac{\pi}{2}, 4)$
* Midline: $(\pi, 3)$
* Min: $(\frac{3\pi}{2}, 2)$
* Midline: $(2\pi, 3)$

4. $y = \cos(x - \pi) + 4$

* Midline: $y = 4$ (Shift up 4).
* Amplitude: $1$. Max is $5$, Min is $3$.
* Phase Shift: Right $\pi$. Standard max is at $0$, so new max is at $\pi$.
* *Note:* Like #2, this is an upside-down cosine relative to the y-axis. It has a minimum at $x=0$.
* Key Points to Plot:
* Min: $(0, 3)$
* Max: $(\pi, 5)$
* Min: $(2\pi, 3)$
* Midlines: $(\frac{\pi}{2}, 4)$ and $(\frac{3\pi}{2}, 4)$

5. $y = -3 \cdot \cos(-\frac{2}{3}x)$

* Simplify: Cosine is an even function, so $\cos(-x) = \cos(x)$. We can treat this as $y = -3 \cos(\frac{2}{3}x)$.
* Midline: $y = 0$ (No vertical shift).
* Amplitude: $3$. Because of the negative sign, the graph is flipped. It starts at a minimum instead of a maximum.
* Max: $3$, Min: $-3$.
* Period: The period is $\frac{2\pi}{B}$. Here $B = \frac{2}{3}$.
* Period $= \frac{2\pi}{2/3} = 2\pi \cdot \frac{3}{2} = 3\pi$.
* Key Points to Plot (One full cycle from $0$ to $3\pi$):
* Start (Min): $(0, -3)$
* Midline (going up): $(\frac{3\pi}{4}, 0)$ *(Quarter period)*
* Max: $(\frac{3\pi}{2}, 3)$ *(Half period)*
* Midline (going down): $(\frac{9\pi}{4}, 0)$ *(Three-quarter period)*
* End (Min): $(3\pi, -3)$ *(Full period)*

6. $y = -2 \cdot \cos(\frac{2}{3}x)$

* Midline: $y = 0$ (No vertical shift).
* Amplitude: $2$. Negative sign means it starts at a minimum.
* Max: $2$, Min: $-2$.
* Period: Same as above, $B = \frac{2}{3}$.
* Period $= 3\pi$.
* Key Points to Plot (One full cycle from $0$ to $3\pi$):
* Start (Min): $(0, -2)$
* Midline (going up): $(\frac{3\pi}{4}, 0)$
* Max: $(\frac{3\pi}{2}, 2)$
* Midline (going down): $(\frac{9\pi}{4}, 0)$
* End (Min): $(3\pi, -2)$

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Final Answer:
To complete the worksheet, draw smooth wave curves connecting the key points calculated above for each graph:

1. Graph 1: Wave oscillating between $y=-1$ and $y=-3$. Peak at $x=-\frac{\pi}{4}$.
2. Graph 2: Wave oscillating between $y=6$ and $y=4$. Valley (min) at $x=0$, Peak at $x=\pi$.
3. Graph 3: Wave oscillating between $y=4$ and $y=2$. Starts at midline $y=3$ at $x=0$ going up. Peak at $x=\frac{\pi}{2}$.
4. Graph 4: Wave oscillating between $y=5$ and $y=3$. Valley (min) at $x=0$, Peak at $x=\pi$.
5. Graph 5: Wave oscillating between $y=3$ and $y=-3$. Starts at bottom ($y=-3$) at $x=0$. Completes one full wave at $x=3\pi$.
6. Graph 6: Wave oscillating between $y=2$ and $y=-2$. Starts at bottom ($y=-2$) at $x=0$. Completes one full wave at $x=3\pi$.