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

x-intercept (y = 0 ) is

The equation of a hyperbola with asymptotes

Cross multiplying and reorganizing,

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

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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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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.

- How do you determine where the given function #f(x) = (x+3)^(2/3) - 6# is concave up and where it is concave down?
- What is the second derivative of #f(x)=x^2/(x^2+3) #?
- For what values of x is #f(x)=((5x)/2)^(2/3) - x^(5/3# concave or convex?
- How do you find intervals where the graph of #f(x) = x + 1/x# is concave up and concave down?
- How do you use the first and second derivatives to sketch #y=(x^3)-(6x^2)+5x+12#?

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