Problem 67 - Algebra and Trigonometry

Question

If you are given the equation of a tangent function, how do you identify an \(x\) -intercept?

Step-by-Step Solution

Answer

The \(x\)-intercepts of a tangent function are determined by setting the function equal to zero and then solving for \(x\). They occur whenever the argument of the tangent function is zero or an even multiple of \(\pi\).

Step 1: Understand the function

The tangent function has a general form of \(y = a \tan(b(x - c))+d\), where \(a\) is the vertical stretch or shrink, \(b\) is the horizontal stretch or shrink, \(c\) is the horizontal shift, and \(d\) is the vertical shift. The \(x\)-intercepts occur wherever the function is equal to zero.

Step 2: Set function to zero

To find the \(x\)-intercepts, we need to set the function equal to zero. So, we take our equation, and set \(y=0\). Solving for \(x\) when \(y=0\) will give us the \(x\)-intercepts. Remember, for any tangent function, the function is zero wherever the argument of the tangent is zero or an even multiple of \(\pi\). This is because the tangent of 0 or an even multiple of \(\pi\) is 0.

Step 3: Solve for \(x\)

Since the function equals zero when the argument of the tangent is zero or an even multiple of \(\pi\), we can solve for \(x\) by setting \(b(x - c) = n\pi\), where \(n\) is any even integer. Hence, solving \(x = \frac{n\pi+b*c}{b}\), will give us the \(x\)-intercepts.