Determine the graph of the functiony = x-3 / x^2 -3x +2
C.
To determine the graph of the function y = (x-3)/(x^2-3x+2), we will follow the steps below:
Step 1: Find the vertical asymptotes.
To find the vertical asymptotes, set the denominator of the function equal to zero and solve for x. We have:
x^2-3x+2 = 0
Factoring the left side, we get:
(x-1)(x-2) = 0
Solving for x, we get:
x = 1 or x = 2
Therefore, the vertical asymptotes are x = 1 and x = 2.
Step 2: Find the horizontal asymptote.
To find the horizontal asymptote, we need to examine the behavior of the function as x approaches positive or negative infinity. We have:
y = (x-3)/(x^2-3x+2)
As x approaches infinity, both the numerator and denominator of the function grow without bounds. Therefore, we can use the ratio of the leading coefficients of the numerator and denominator to find the horizontal asymptote. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 1, so the horizontal asymptote is y = 0.
Step 3: Find the x-intercepts.
To find the x-intercepts, we need to set the numerator of the function equal to zero and solve for x. We have:
x-3 = 0
Solving for x, we get:
x = 3
Therefore, the x-intercept is (3, 0).
Step 4: Find the y-intercept.
To find the y-intercept, we need to set x equal to zero in the function. We have:
y = (-3)/(2)
Therefore, the y-intercept is (0, -3/2).
Step 5: Determine the behavior of the function near the vertical asymptotes.
To determine the behavior of the function near the vertical asymptotes, we need to examine the signs of the numerator and denominator of the function as x approaches the vertical asymptotes from both sides. We have:
For x < 1: (x-3) is negative and (x^2-3x+2) is negative, so y is positive.
For 1 < x < 2: (x-3) is negative and (x^2-3x+2) is positive, so y is negative.
For x > 2: (x-3) is positive and (x^2-3x+2) is positive, so y is positive.
Therefore, the graph of the function approaches the vertical asymptotes from both sides.
Step 6: Sketch the graph.
Using all the information gathered in the previous steps, we can sketch the graph of the function as shown below:
graph{y=(x-3)/(x^2-3x+2) [-10, 10, -10, 10]}
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