Pregunta
Given:
8.1 Give the equations of the asymptotes of
dind
8.2 Determine the
-and
-intercepts of the graph of
.
8.3 Sketch the graph of
using the grid on the DIAGRAM SHEET
8.4 Give the domain of
.
8.5 Describe the transformation of
to
if:
8.5.1
8.1 Give the equations of the asymptotes of
8.2 Determine the
8.3 Sketch the graph of
8.4 Give the domain of
8.5 Describe the transformation of
8.5.1
Ask by Johnson Cervantes. in South Africa
Mar 15,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
8.1 Asymptotes
- Vertical asymptote:
- Horizontal asymptote:
8.2 Intercepts
-
-intercept: -
-intercept:
8.3 Graph Sketch
- Plot vertical asymptote at
and horizontal asymptote at . - The graph has two branches:
- For
: approaches as approaches 2 from the left. - For
: approaches as approaches 2 from the right.
- For
- As
, approaches 1. - Plot the intercept at
and sketch the graph accordingly.
8.4 Domain
All real numbers except
:
8.5 Transformations
-
8.5.1
:
Shifts the graph ofleft by 3 units. - New vertical asymptote:
- Horizontal asymptote remains
- New vertical asymptote:
-
8.5.2
:
Shifts the graph ofdown by 2 units. - New horizontal asymptote:
- Vertical asymptote remains
- New horizontal asymptote:
Solución
8.1 Asymptotes
- Vertical asymptote: The function is undefined when the denominator is zero, i.e., when
- Horizontal asymptote: When
becomes very large (or very small), tends to zero so that Thus, the horizontal asymptote is .
8.2
- and
-intercepts
-
-intercept: Set and solve for : Subtract 1 from both sides:Multiply both sides by(noting ): Simplify the right-hand side:Subtract 2 from both sides:soThe-intercept is at . -
-intercept: Substitute into : The-intercept is also at .
8.3 Sketching the Graph
- Plot the asymptotes: the vertical line
and the horizontal line . - The intercept of the graph is at
. - The graph will have two branches separated by the vertical asymptote
: - For
: as approaches 2 from the left, goes to . - For
: as approaches 2 from the right, goes to .
- For
- As
, .
Plot these features on the coordinate grid.
8.4 Domain
The function
is defined for all real numbers except where the denominator is zero. Thus, the domain is:
8.5 Transformations
-
8.5.1 For
:
This represents a horizontal shift of the graph ofto the left by units. - The vertical asymptote shifts from
to which gives . - The horizontal asymptote remains
since horizontal shifts do not affect it.
- The vertical asymptote shifts from
-
8.5.2 For
:
This represents a vertical shift of the graph ofdownward by units. - The horizontal asymptote shifts from
to . - The vertical asymptote remains
because vertical shifts do not affect it.
- The horizontal asymptote shifts from
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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Mind Expander
The equation
reveals that there are two important asymptotes: a vertical asymptote at
where the function approaches infinity and a horizontal asymptote at
, as the term
vanishes when
is very large or very small. These asymptotes indicate the boundaries of the graph’s behavior as it stretches toward the edges of the coordinate plane.
When determining the intercepts of
, the
-intercept can be found by substituting
into the equation, which gives
. Thus, the
-intercept is at the point
. For the
-intercept, you set
leading to
, which simplifies to
, giving the point
.

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