AP Precalculus Practice Test: Unit 3 Question #28 Find the Vertical Asymptotes of y = 3tan(x/2) + 6

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**AP Precalculus Practice Test: Unit 3, Question #28** asks you to find the vertical asymptotes of the function \( y = 3 \tan\left(\frac{x}{2}\right) + 6 \). For tangent functions, vertical asymptotes occur at the values of \(x\) where the function is undefined due to division by zero in the tangent formula.

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### **Key Concepts**
1. The general tangent function \( y = a \tan(bx - h) + k \) has vertical asymptotes where:
\[
bx - h = \frac{\pi}{2} + n\pi
\]
where \(n\) is any integer.

2. The given function \( y = 3 \tan\left(\frac{x}{2}\right) + 6 \) has:
- \( b = \frac{1}{2} \)
- No horizontal shift (\( h = 0 \))
- A vertical shift of \( k = 6 \), which does not affect the vertical asymptotes.

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### **Finding the Vertical Asymptotes**
1. Start with the equation for vertical asymptotes of the tangent function:
\[
\frac{x}{2} = \frac{\pi}{2} + n\pi
\]

2. Solve for \(x\) by multiplying through by 2:
\[
x = \pi + 2n\pi
\]

3. Simplify the expression:
\[
x = \pi(1 + 2n)
\]
where \(n\) is any integer.

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### **Description**:
"The vertical asymptotes of the function occur at all \(x\)-values where \(x = \pi(1 + 2n)\), with \(n\) representing any integer. These values correspond to the points where the tangent function becomes undefined."

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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