Sketch the following curves
Y=x/x^2+1
Y=(x+2)(x-3)/x+1
First curve: Y = x/(x^2 + 1)
Let's start by finding the intercepts of the curve. To find the x-intercept, we set y = 0:
0 = x/(x^2 + 1)
Since the numerator cannot be equal to zero (x ≠ 0), we can disregard it. So, x^2 + 1 = 0, which has no real solutions. Therefore, there is no x-intercept.
To find the y-intercept, we set x = 0:
y = 0/(0^2 + 1)
y = 0/1
y = 0
The y-intercept is at (0, 0).
Next, we analyze the behavior of the curve as x approaches positive and negative infinity. Consider the limit of y as x approaches infinity:
lim(x→∞) x/(x^2 + 1)
This limit can be determined by comparing the degrees of the numerator and denominator. The degree of the denominator (x^2) is greater than the degree of the numerator (x), so as x approaches infinity, the fraction gets closer and closer to 0. Therefore, the curve approaches the x-axis as x goes to infinity.
Similarly, as x approaches negative infinity (-∞):
lim(x→-∞) x/(x^2 + 1)
Again, we compare the degrees of the numerator and denominator. The degree of the denominator (x^2) is greater than the degree of the numerator (x), so as x approaches negative infinity, the fraction gets closer and closer to 0. Therefore, the curve approaches the x-axis as x goes to negative infinity.
Now, let's find any vertical asymptotes. These occur when the denominator equals zero (x^2 + 1 = 0). However, this equation has no real solutions, so there are no vertical asymptotes.
For horizontal asymptotes, we look at the behavior of the curve as x approaches positive and negative infinity. As mentioned before, the curve approaches the x-axis in both cases. Therefore, the horizontal asymptote is y = 0.
Based on the above analysis, we can sketch the curve as follows:
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-4 -2 0 2 4
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Second curve: Y = (x + 2)(x - 3)/(x + 1)
Again, let's find the intercepts. To find the x-intercept, we set y = 0:
0 = (x + 2)(x - 3)/(x + 1)
Setting each factor equal to zero, we have:
x + 2 = 0, x - 3 = 0
So, x = -2 and x = 3. The x-intercepts are (-2, 0) and (3, 0).
To find the y-intercept, we set x = 0:
y = (0 + 2)(0 - 3)/(0 + 1)
y = (-2)(-3)/1
y = 6
The y-intercept is at (0, 6).
Now, let's analyze the behavior as x approaches positive and negative infinity. Consider the limit of y as x approaches infinity:
lim(x→∞)(x + 2)(x - 3)/(x + 1)
We compare the degrees of the numerator and denominator. The degree of the numerator (x^2) is the same as the degree of the denominator (x^1), so we divide the coefficients of the highest degree terms, which is 1/1 = 1. Therefore, the curve approaches a horizontal asymptote at y = 1 as x goes to infinity.
As x approaches negative infinity (-∞):
lim(x→-∞)(x + 2)(x - 3)/(x + 1)
Again, we compare the degrees and divide the coefficients of the highest degree terms (1/1 = 1). The curve approaches a horizontal asymptote at y = 1 as x goes to negative infinity.
Next, we look for vertical asymptotes by setting the denominator equal to zero (x + 1 = 0). This gives us x = -1, so there is a vertical asymptote at x = -1.
Based on the above analysis, we can sketch the curve as follows:
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-4 -2 0 2 4
Keep in mind that these are just rough sketches based on the analysis provided. For more accuracy, additional points and a connection between them would be necessary.