Question
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
.
Given the function:
5.2 Draw the graph of
5.3 Determine the equation of the axis of symmetry of
Ask by Love Bryant. in South Africa
Nov 14,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
- 5.1: Vertical asymptote:
; Horizontal asymptote: . - 5.2: Graph includes intercepts at
and , with asymptotes drawn. - 5.3: Axis of symmetry:
.
- 5.2: Graph includes intercepts at
Solution

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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
.