Math Problem Statement

Identify the function whose graph appears above.

Solution

The graph you provided shows a set of vertical asymptotes at regular intervals, specifically at integer values of xx where x=±2,±4,±6,x = \pm 2, \pm 4, \pm 6, etc. Additionally, the shape of the curves between these asymptotes suggests the presence of repeated cycles, characteristic of a tangent or cotangent function with periodic asymptotes.

Given these features, the function likely takes the form: f(x)=tan(π2x) or f(x)=tan(π2x)f(x) = \tan\left(\frac{\pi}{2} x\right) \text{ or } f(x) = -\tan\left(\frac{\pi}{2} x\right) depending on whether the curve opens upward or downward between asymptotes. In this case, the curves open downward on both sides of each vertical asymptote, suggesting the function is of the form: f(x)=tan(π2x)f(x) = -\tan\left(\frac{\pi}{2} x\right)

Final Answer:

f(x)=tan(π2x)f(x) = -\tan\left(\frac{\pi}{2} x\right)

Would you like further details or have any questions?

Here are five questions to expand on this:

  1. How can you determine the period of a trigonometric function from its graph?
  2. What are vertical asymptotes, and why do they appear in tangent functions?
  3. How does the transformation tan(π2x)-\tan\left(\frac{\pi}{2} x\right) differ from tan(π2x)\tan\left(\frac{\pi}{2} x\right)?
  4. How can one identify whether a graph represents a tangent or cotangent function?
  5. What are the characteristics of the tangent function’s graph, such as symmetry and asymptotes?

Tip: When identifying trigonometric functions, pay close attention to asymptotes and intervals—they provide essential clues about the function's periodicity and transformations.

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Math Problem Analysis

Mathematical Concepts

Trigonometric Functions
Tangent Function
Graph Analysis

Formulas

f(x) = -tan(π/2 * x)

Theorems

Properties of Tangent Function

Suitable Grade Level

Grades 11-12