Sketch the graph of the function
f(x) = (x+4)3(x+1)(x−2)2(x−5)/
(x+3)2(x+1)(x−3)
vertical asymptotes at -3,3
zeroes at -4,2,5
multiplicity 3,2,1
hole at x = -1 (x+1) occurs in both top and bottom
no horizontal asymptotes
http://www.wolframalpha.com/input/?i=((x%2B4)%5E3(x%2B1)(x%E2%88%922)%5E2(x%E2%88%925))%2F((x%2B3)%5E2(x%2B1)(x%E2%88%923))+for+-5%3C%3Dx%3C%3D6
To sketch the graph of the given function, we can follow these steps:
1. Determine the x-intercepts:
To find the x-intercepts, set the numerator equal to zero and solve:
(x + 4)3(x + 1)(x - 2)2(x - 5) = 0
This gives us the x-intercepts at x = -4, x = -1, x = 2, and x = 5.
2. Determine the y-intercept:
To find the y-intercept, set x = 0 and evaluate the function:
f(0) = (0 + 4)3(0 + 1)(0 - 2)2(0 - 5) / (0 + 3)2(0 + 1)(0 - 3)
Simplify this expression to find the y-intercept.
3. Analyze the behavior for large x values:
As x approaches positive or negative infinity, the function will approach a horizontal asymptote. Determine the asymptote by examining the degree of the highest power of x in the numerator and denominator. In this case, since the degree of the numerator is higher than the denominator, there is no horizontal asymptote.
4. Determine the behavior around vertical asymptotes:
To find the vertical asymptotes, set the denominator equal to zero and solve for the values of x that make the denominator zero. In this case, we find vertical asymptotes at x = -3 and x = 3.
5. Determine the behavior around x = -1 and x = 1:
Since the factors (x + 1) and (x - 2) have odd powers in the numerator, the function will cross the x-axis at x = -1 and x = 2.
6. Determine the behavior around x = 5:
Since the factor (x - 5) has an even power in the numerator, the function will touch the x-axis but not cross it at x = 5.
7. Plot the intercepts, asymptotes, and note the behavior:
Based on the information gathered from the above steps, plot the x-intercepts, y-intercept, vertical asymptotes, and note any behavior around the other critical points.
By following these steps, you can sketch the graph of the given function.