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QUESTION 5
Given the function: .
Write down the equations of the asymptotes of .
5.2 Draw the graph of . Clearly show all asymptotes and intercepts with the axes.
5.3 Determine the equation of the axis of symmetry of with .

Ask by Love Bryant. in South Africa
Nov 14,2024

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Answer

  • 5.1: Vertical asymptote: ; Horizontal asymptote: .
    • 5.2: Graph includes intercepts at and , with asymptotes drawn.
    • 5.3: Axis of symmetry: .

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Beyond the Answer

The function has both vertical and horizontal asymptotes. The vertical asymptote occurs where the denominator equals zero, which is at . The horizontal asymptote is determined by analyzing the end behavior of the function as approaches infinity, leading to (as the term involving vanishes). Thus, the asymptotes are and .
When graphing , start by marking the vertical asymptote at and the horizontal asymptote at . The function approaches these asymptotes but never touches them. The x-intercept can be found by setting , giving us , leading to . The y-intercept can be found by evaluating , giving . The graph will show a hyperbolic shape, approaching both asymptotes as it moves away from the intercepts.
The axis of symmetry for the function can be determined by examining the properties of the rational function. In this case, the axis of symmetry is vertical because the function is not even. A function can be considered symmetric about a vertical line if for some . Here, it simplifies to , which is quite standard for rational functions resembling hyperbolas, thus indicating reflection symmetry in relation to where .

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