Pregunta
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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

Ask by Johnson Cervantes. in South Africa
Mar 15,2025

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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.
  • 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 of left by 3 units.
    • New vertical asymptote:
    • Horizontal asymptote remains
  • 8.5.2 :
    Shifts the graph of down by 2 units.
    • New horizontal asymptote:
    • Vertical asymptote remains

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:
    so
    The -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 .
  • 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 of to the left by units.
    • The vertical asymptote shifts from to which gives .
    • The horizontal asymptote remains since horizontal shifts do not affect it.
  • 8.5.2 For :
    This represents a vertical shift of the graph of downward by units.
    • The horizontal asymptote shifts from to .
    • The vertical asymptote remains because vertical shifts do not affect it.

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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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