How do you find intercepts, extrema, points of inflections, asymptotes and graph #y=(x+2)/x#?

Answer 1

x-intercept (y = 0 ) is #-2#. Asymptotes are horizontal #larr y =1 rarr and vertical uarr x = 0 darr#. No extrema. No point of inflexion. Graph is inserted.

The equation of a hyperbola with asymptotes

#y= m_1x+c_1 and y = m_2x+c_2# is
#(y-m_1x-c_1)(y-m_2x-c_2)=# non-zero constant.

Cross multiplying and reorganizing,

#x(y-1)=-2#. So,

the asymptotes are x = 0 and y =1 1 that are at right angles.

As # y in (-oo, oo)#, sans 1, there are no extrema.
#y''=4/x^3# that cannot become 0. So, there is no point of inflexion. graph{x(y-1)-2=0 [-10, 10, -5, 5]}
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Answer 2

To find intercepts:

  • x-intercept: Set y = 0 and solve for x.
  • y-intercept: Set x = 0 and solve for y.

To find extrema:

  • Take the derivative of the function.
  • Set the derivative equal to zero and solve for x.
  • These x-values represent potential extrema. Test each value to determine if it corresponds to a maximum, minimum, or neither.

To find points of inflection:

  • Take the second derivative of the function.
  • Set the second derivative equal to zero and solve for x.
  • These x-values represent potential points of inflection. Test each value to confirm.

To find asymptotes:

  • Vertical asymptotes: Set the denominator equal to zero and solve for x. These values represent vertical asymptotes.
  • Horizontal asymptote: Determine the behavior of the function as x approaches positive and negative infinity. If the function approaches a constant value, that's the horizontal asymptote.

Graphing the function:

  • Plot the intercepts, extrema, points of inflection, and asymptotes on the graph.
  • Sketch the curve, ensuring it adheres to the characteristics found above.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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