How do you graph #y =(x^2+4x-2)/(x^2-x-7)#?

Answer 1

See graph and details.

By actual division,

#y = 1 + ( 5 ( x + 1 ))/(( x - a )(x - b )), a,b = 1/2( 1 +- sqrt29 )#. So,
#( y - 1 )( x - a )( x - b ) = 5 ( x + 1 )#, with the asymptotic
#y ne 1, x ne a and x ne b )#.

See graph, with asymptotes. y = 1, x = a and x = b. graph{(y(x^2-x-7)-(x^2+4x-2))(y-1)(x-1/2-1/2sqrt29+0.00001y)(x-1/2+1/2sqrt29+0.00001y)=0}

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Answer 2

To graph the function y = (x^2 + 4x - 2)/(x^2 - x - 7), follow these steps:

  1. Determine the domain of the function by finding the values of x for which the denominator (x^2 - x - 7) is not equal to zero. In this case, the denominator factors as (x - 3)(x + 2), so the function is undefined at x = 3 and x = -2.

  2. Find the y-intercept by substituting x = 0 into the equation. y = (0^2 + 4(0) - 2)/(0^2 - 0 - 7) simplifies to y = -2/(-7) = 2/7.

  3. Determine the x-intercepts by setting y = 0 and solving for x. However, in this case, the numerator (x^2 + 4x - 2) does not factor easily. Therefore, you can use a graphing calculator or software to find the approximate x-intercepts.

  4. Analyze the behavior of the function as x approaches positive and negative infinity. As x becomes very large or very small, the terms involving x^2 dominate the function. Therefore, the function approaches y = 1 as x approaches positive or negative infinity.

  5. Plot the obtained points, including the y-intercept and x-intercepts, on a graph. Use the behavior analysis to sketch the curve between the intercepts.

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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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