Trig graph transformations worksheet - Free Printable

To solve the problem of graphing the given cosine functions, we need to understand how transformations affect the standard cosine function \( y = \cos(x) \). The general form of a transformed cosine function is:

\[
y = A \cos(Bx - C) + D
\]

Where:
- \( A \) is the amplitude (vertical stretch or compression).
- \( B \) affects the period (horizontal stretch or compression).
- \( C \) causes a horizontal shift (phase shift).
- \( D \) causes a vertical shift.

Let's go through each function step by step.

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

#### Analysis:
- Amplitude: \( A = 1 \) (no vertical stretch or compression).
- Period: \( B = 1 \), so the period is \( \frac{2\pi}{B} = 2\pi \).
- Phase Shift: \( C = \frac{\pi}{4} \), so the phase shift is \( -\frac{C}{B} = -\frac{\pi}{4} \) (shift left by \( \frac{\pi}{4} \)).
- Vertical Shift: \( D = -2 \), so the graph is shifted down by 2 units.

#### Steps to Graph:
1. Start with the standard cosine function \( y = \cos(x) \).
2. Shift the graph left by \( \frac{\pi}{4} \).
3. Shift the graph down by 2 units.

#### Key Points:
- At \( x = -\frac{\pi}{4} \), \( y = \cos(0) - 2 = 1 - 2 = -1 \).
- At \( x = \frac{3\pi}{4} \), \( y = \cos(\pi) - 2 = -1 - 2 = -3 \).
- At \( x = \frac{7\pi}{4} \), \( y = \cos(2\pi) - 2 = 1 - 2 = -1 \).

#### Graph:
Plot these points and sketch the cosine curve with the specified shifts.

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

#### Analysis:
- Amplitude: \( A = 1 \).
- Period: \( B = 1 \), so the period is \( 2\pi \).
- Phase Shift: \( C = \pi \), so the phase shift is \( -\frac{\pi}{1} = -\pi \) (shift left by \( \pi \)).
- Vertical Shift: \( D = 5 \), so the graph is shifted up by 5 units.

#### Steps to Graph:
1. Start with the standard cosine function \( y = \cos(x) \).
2. Shift the graph left by \( \pi \).
3. Shift the graph up by 5 units.

#### Key Points:
- At \( x = -\pi \), \( y = \cos(0) + 5 = 1 + 5 = 6 \).
- At \( x = 0 \), \( y = \cos(\pi) + 5 = -1 + 5 = 4 \).
- At \( x = \pi \), \( y = \cos(2\pi) + 5 = 1 + 5 = 6 \).

#### Graph:
Plot these points and sketch the cosine curve with the specified shifts.

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

#### Analysis:
- Amplitude: \( A = 1 \).
- Period: \( B = 1 \), so the period is \( 2\pi \).
- Phase Shift: \( C = \frac{\pi}{2} \), so the phase shift is \( -\frac{-\pi/2}{1} = \frac{\pi}{2} \) (shift right by \( \frac{\pi}{2} \)).
- Vertical Shift: \( D = 3 \), so the graph is shifted up by 3 units.

#### Steps to Graph:
1. Start with the standard cosine function \( y = \cos(x) \).
2. Shift the graph right by \( \frac{\pi}{2} \).
3. Shift the graph up by 3 units.

#### Key Points:
- At \( x = \frac{\pi}{2} \), \( y = \cos(0) + 3 = 1 + 3 = 4 \).
- At \( x = \frac{3\pi}{2} \), \( y = \cos(\pi) + 3 = -1 + 3 = 2 \).
- At \( x = \frac{5\pi}{2} \), \( y = \cos(2\pi) + 3 = 1 + 3 = 4 \).

#### Graph:
Plot these points and sketch the cosine curve with the specified shifts.

---

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

#### Analysis:
- Amplitude: \( A = 1 \).
- Period: \( B = 1 \), so the period is \( 2\pi \).
- Phase Shift: \( C = \pi \), so the phase shift is \( -\frac{-\pi}{1} = \pi \) (shift right by \( \pi \)).
- Vertical Shift: \( D = 4 \), so the graph is shifted up by 4 units.

#### Steps to Graph:
1. Start with the standard cosine function \( y = \cos(x) \).
2. Shift the graph right by \( \pi \).
3. Shift the graph up by 4 units.

#### Key Points:
- At \( x = \pi \), \( y = \cos(0) + 4 = 1 + 4 = 5 \).
- At \( x = 2\pi \), \( y = \cos(\pi) + 4 = -1 + 4 = 3 \).
- At \( x = 3\pi \), \( y = \cos(2\pi) + 4 = 1 + 4 = 5 \).

#### Graph:
Plot these points and sketch the cosine curve with the specified shifts.

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

#### Analysis:
- Amplitude: \( A = 3 \) (vertical stretch by 3 and reflection over the x-axis).
- Period: \( B = \frac{2}{3} \), so the period is \( \frac{2\pi}{\frac{2}{3}} = 3\pi \).
- Phase Shift: \( C = 0 \), so no phase shift.
- Vertical Shift: \( D = 0 \), so no vertical shift.

#### Steps to Graph:
1. Start with the standard cosine function \( y = \cos(x) \).
2. Reflect the graph over the x-axis.
3. Stretch the graph vertically by a factor of 3.
4. Compress the graph horizontally by a factor of \( \frac{2}{3} \) (period becomes \( 3\pi \)).

#### Key Points:
- At \( x = 0 \), \( y = -3 \cdot \cos(0) = -3 \cdot 1 = -3 \).
- At \( x = \frac{3\pi}{2} \), \( y = -3 \cdot \cos\left(-\frac{2}{3} \cdot \frac{3\pi}{2}\right) = -3 \cdot \cos(-\pi) = -3 \cdot (-1) = 3 \).
- At \( x = 3\pi \), \( y = -3 \cdot \cos\left(-\frac{2}{3} \cdot 3\pi\right) = -3 \cdot \cos(-2\pi) = -3 \cdot 1 = -3 \).

#### Graph:
Plot these points and sketch the cosine curve with the specified transformations.

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

#### Analysis:
- Amplitude: \( A = 2 \) (vertical stretch by 2 and reflection over the x-axis).
- Period: \( B = \frac{2}{3} \), so the period is \( \frac{2\pi}{\frac{2}{3}} = 3\pi \).
- Phase Shift: \( C = 0 \), so no phase shift.
- Vertical Shift: \( D = 0 \), so no vertical shift.

#### Steps to Graph:
1. Start with the standard cosine function \( y = \cos(x) \).
2. Reflect the graph over the x-axis.
3. Stretch the graph vertically by a factor of 2.
4. Compress the graph horizontally by a factor of \( \frac{2}{3} \) (period becomes \( 3\pi \)).

#### Key Points:
- At \( x = 0 \), \( y = -2 \cdot \cos(0) = -2 \cdot 1 = -2 \).
- At \( x = \frac{3\pi}{2} \), \( y = -2 \cdot \cos\left(\frac{2}{3} \cdot \frac{3\pi}{2}\right) = -2 \cdot \cos(\pi) = -2 \cdot (-1) = 2 \).
- At \( x = 3\pi \), \( y = -2 \cdot \cos\left(\frac{2}{3} \cdot 3\pi\right) = -2 \cdot \cos(2\pi) = -2 \cdot 1 = -2 \).

#### Graph:
Plot these points and sketch the cosine curve with the specified transformations.

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Final Answer:

\[
\boxed{
\text{Graph each function as described above.}
}
\]