Trigonometric Graphs and their transformations - Free Printable

To solve this worksheet, we need to graph each trigonometric equation on the provided coordinate planes. The domain for all graphs is from $-2\pi$ to $2\pi$.

Here is the step-by-step breakdown for how to draw each graph:

1. $y = \sin x$

* Type: Standard Sine Wave.
* Key Points: Passes through $(0,0)$. Goes up to $1$ at $\frac{\pi}{2}$, down through $0$ at $\pi$, down to $-1$ at $\frac{3\pi}{2}$, and back to $0$ at $2\pi$.
* Negative Side: Since sine is an odd function, it goes down to $-1$ at $-\frac{\pi}{2}$ and up to $1$ at $-\frac{3\pi}{2}$.
* Shape: An "S" shape curve centered at the origin.

2. $y = \cos x$

* Type: Standard Cosine Wave.
* Key Points: Starts at maximum height $(0,1)$. Goes down to $0$ at $\frac{\pi}{2}$, minimum $-1$ at $\pi$, up to $0$ at $\frac{3\pi}{2}$, and max $1$ at $2\pi$.
* Negative Side: It is symmetric across the y-axis. It looks exactly the same on the left side as the right side.
* Shape: A "U" shaped valley centered at the y-axis (but rounded).

3. $y = 2 \sin x$

* Transformation: Vertical Stretch by a factor of 2.
* Amplitude: The graph goes twice as high and low as #1.
* Key Points: Max is $2$ (at $\frac{\pi}{2}$), Min is $-2$ (at $\frac{3\pi}{2}$). It still crosses zero at $0, \pi, 2\pi$, etc.
* Shape: Same "S" shape as #1, but taller.

4. $y = 3 \cos x$

* Transformation: Vertical Stretch by a factor of 3.
* Amplitude: The graph goes three times as high and low as #2.
* Key Points: Max is $3$ (at $0, 2\pi$), Min is $-3$ (at $\pi$).
* Shape: Same wave shape as #2, but much taller.

5. $y = -\sin x$

* Transformation: Reflection across the x-axis (flipped upside down).
* Key Points: Starts at $(0,0)$. Instead of going up first, it goes down to $-1$ at $\frac{\pi}{2}$. It goes up to $1$ at $\frac{3\pi}{2}$.
* Shape: Looks like #1 but flipped vertically.

6. $y = -2 \sin x$

* Transformation: Reflection AND Vertical Stretch by 2.
* Key Points: Starts at $(0,0)$. Goes down to $-2$ at $\frac{\pi}{2}$. Goes up to $2$ at $\frac{3\pi}{2}$.
* Shape: Looks like #3 but flipped upside down.

7. $y = \frac{1}{2} \sin(x - \frac{\pi}{2})$

* Transformation: This is a phase shift. Subtracting $\frac{\pi}{2}$ inside the parenthesis shifts the graph to the right by $\frac{\pi}{2}$.
* Shortcut: Shifting a sine wave right by $\frac{\pi}{2}$ makes it look exactly like a negative cosine wave. So, $y = -\frac{1}{2} \cos x$.
* Key Points: At $x=0$, $y = -\frac{1}{2}$. At $x=\frac{\pi}{2}$, $y=0$. At $x=\pi$, $y=\frac{1}{2}$ (peak). At $x=\frac{3\pi}{2}$, $y=0$.
* Amplitude: Only goes up to $0.5$ and down to $-0.5$.

8. $y = \sin(2x - \pi) + 1$

* Transformations:
1. Period Change: The "$2$" in front of $x$ means the wave moves twice as fast. The period is $\pi$ instead of $2\pi$.
2. Phase Shift: Factor out the 2: $\sin(2(x - \frac{\pi}{2}))$. This shifts the graph right by $\frac{\pi}{2}$.
3. Vertical Shift: The "$+1$" at the end moves the whole graph up by 1 unit. The midline is now $y=1$.
* Key Points:
* At $x=0$: $\sin(-\pi)+1 = 0+1 = 1$. Point: $(0,1)$.
* At $x=\frac{\pi}{2}$: $\sin(0)+1 = 1$. Point: $(\frac{\pi}{2}, 1)$.
* At $x=\frac{3\pi}{4}$: $\sin(\frac{\pi}{2})+1 = 2$. Peak at $(\frac{3\pi}{4}, 2)$.
* At $x=\frac{\pi}{4}$: $\sin(-\frac{\pi}{2})+1 = 0$. Trough at $(\frac{\pi}{4}, 0)$.
* Shape: A wave oscillating between $y=0$ and $y=2$, centered on the line $y=1$.

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Final Answer:
The solutions are the graphs described above. Here is a summary of the key features for plotting:

1. $y = \sin x$: Standard sine wave, amplitude 1, passes through origin.
2. $y = \cos x$: Standard cosine wave, amplitude 1, starts at peak $(0,1)$.
3. $y = 2 \sin x$: Sine wave stretched vertically; peaks at $2$, troughs at $-2$.
4. $y = 3 \cos x$: Cosine wave stretched vertically; peaks at $3$, troughs at $-3$.
5. $y = -\sin x$: Sine wave flipped upside down; goes down first from origin.
6. $y = -2 \sin x$: Sine wave flipped and stretched; goes down to $-2$ first.
7. $y = \frac{1}{2} \sin(x - \frac{\pi}{2})$: Small wave (height $0.5$); starts at $-0.5$ on the y-axis (looks like inverted cosine).
8. $y = \sin(2x - \pi) + 1$: Wave shifted up by 1 (oscillates between 0 and 2); completes two full cycles between $0$ and $2\pi$.