How do you graph #f(x)=(2x-3)/(x^2+x-6)# using holes, vertical and horizontal asymptotes, x and y intercepts?
Below
Plotting your intercepts and drawing in your asymptotes, you should get something like below. Remember, your asymptotes only affect/influence the endpoints of your graph and not anywhere else on your graph
graph{(2x-3)/(x^2+x-6) [-10, 10, -5, 5]}
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To graph the function f(x) = (2x-3)/(x^2+x-6), we can follow these steps:
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Factorize the denominator (x^2+x-6) to find any potential holes or vertical asymptotes: x^2 + x - 6 = (x-2)(x+3) Therefore, the function has potential holes at x = 2 and x = -3.
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Determine the vertical asymptotes by finding the values of x that make the denominator equal to zero: x^2 + x - 6 = 0 Solving this quadratic equation, we find x = 2 and x = -3. Hence, the vertical asymptotes are x = 2 and x = -3.
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Find the x-intercepts by setting the numerator equal to zero and solving for x: 2x - 3 = 0 Solving this equation, we get x = 3/2. Therefore, the x-intercept is (3/2, 0).
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Find the y-intercept by evaluating the function at x = 0: f(0) = (2(0) - 3)/(0^2 + 0 - 6) = -3/6 = -1/2. Hence, the y-intercept is (0, -1/2).
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Determine the behavior of the function as x approaches positive or negative infinity to find the horizontal asymptotes: As x approaches positive or negative infinity, the function approaches zero. Therefore, the horizontal asymptote is y = 0.
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Plot the obtained points: holes at x = 2 and x = -3, vertical asymptotes at x = 2 and x = -3, x-intercept at (3/2, 0), y-intercept at (0, -1/2), and the horizontal asymptote at y = 0.
By following these steps, you can graph the function f(x) = (2x-3)/(x^2+x-6) accurately.
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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.
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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