To create a cosine function with a specific period, you use the period formula for trigonometric functions:

$\text{Period} = \frac{2\pi}{B},$

where $B$ is the coefficient of $x$ inside the cosine function $\cos(Bx)$. To create a function with a desired period, you solve for $B$ using:

$B = \frac{2\pi}{\text{Period}}.$

Steps to Create a Cosine Function with a Given Period:

1. Determine the desired period $T$.
2. Compute $B = \frac{2\pi}{T}$.
3. Write the function in the form $y = \cos(Bx)$.

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Example:

If the desired period is $4$:

1. Desired period $T = 4$.
2. Compute $B = \frac{2\pi}{4} = \frac{\pi}{2}$.
3. The cosine function becomes: $y = \cos\left(\frac{\pi}{2}x\right).$

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General Form with Adjustments:

To further modify the cosine function:

• Amplitude: Multiply $\cos(Bx)$ by a factor $A$ (e.g., $A \cos(Bx)$).
• Vertical shift: Add or subtract a constant $D$ (e.g., $A \cos(Bx) + D$).
• Horizontal shift: Add a phase shift $C$ inside the cosine (e.g., $\cos(B(x - C))$).

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Would you like more examples or further explanations?

Related Questions:

1. What is the difference between period and amplitude in trigonometric functions?
2. How do you graph a cosine function with a specific period and amplitude?
3. How do you determine the period from a cosine equation like $y = \cos(3x)$?
4. What are real-life applications of periodic cosine functions?
5. How does a vertical shift affect the graph of a cosine function?

Tip:

Always double-check the period by recalculating $T = \frac{2\pi}{B}$ after determining $B$.