Graph of Arctangent

Definition

The graph of arctangent, or the inverse tangent function, is a visual representation of the values of the function $$y = \tan^{-1}(x)$$, which indicates the angle whose tangent is x. It is defined for all real numbers and has a characteristic S-shape, with horizontal asymptotes at $$y = \frac{-\pi}{2}$$ and $$y = \frac{\pi}{2}$$. The graph demonstrates how the arctangent function maps inputs from the entire x-axis to outputs between these two asymptotes.

5 Must Know Facts For Your Next Test

1. The graph of arctangent passes through the origin, meaning that when $$x = 0$$, $$y = 0$$.
2. As $$x$$ approaches positive infinity, the value of $$y$$ approaches $$\frac{\pi}{2}$$, and as $$x$$ approaches negative infinity, $$y$$ approaches $$\frac{-\pi}{2}$$.
3. The graph is increasing throughout its domain, indicating that it is a one-to-one function and has an inverse.
4. The slope of the graph of arctangent is steepest at $$x = 0$$ and gradually decreases as you move away from this point.
5. The range of the arctangent function is limited to values between $$\frac{-\pi}{2}$$ and $$\frac{\pi}{2}$$.

Review Questions

• Explain how the graph of arctangent demonstrates the relationship between input values and their corresponding angles.
  • The graph of arctangent shows how each real number input corresponds to an angle whose tangent equals that number. As you move along the x-axis, each value produces a unique output angle between $$\frac{-\pi}{2}$$ and $$\frac{\pi}{2}$$. The continuous increase in y-values illustrates that every input has exactly one output, confirming that it is a one-to-one function.
• How do the asymptotes in the graph of arctangent affect its shape and behavior?
  • The asymptotes in the graph of arctangent at $$y = \frac{-\pi}{2}$$ and $$y = \frac{\pi}{2}$$ create boundaries that limit the output values of the function. As you approach these asymptotes, the graph gets closer to these lines but never actually touches them. This S-shaped curve reflects that while the function continues indefinitely along the x-axis, its outputs remain confined within this finite range.
• Analyze how understanding the graph of arctangent can be applied to solve problems involving angles in right triangles.
  • Understanding the graph of arctangent enables students to find angles when given tangent ratios in right triangle problems. For instance, if you know that the tangent of an angle is 1, you can locate this value on the x-axis and see that it corresponds to an angle of $$\frac{\pi}{4}$$ radians. This application highlights how inverse functions can be used practically in trigonometry by linking numerical ratios back to angles.

Related terms

Tangent Function: The tangent function, defined as $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$, describes the ratio of the lengths of the opposite side to the adjacent side in a right triangle.

Inverse Functions: Inverse functions are functions that reverse the effect of the original function, meaning if $$f(x)$$ gives you y, then $$f^{-1}(y)$$ gives you x.

Asymptotes: Asymptotes are lines that a graph approaches but never touches, indicating the behavior of a function as it reaches extreme values.