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Given the function , use the characteristics of polynomials and rational functions to describe its behavior and sketch the function.Enter the exact answers.Enter the intercepts as points, . Enter the -intercepts in increasing order of the -coordinate.The -intercepts are and The -intercept is The fields below accept a list of numbers or formulas separated by semicolons (e.g. or . The order of the list does not matter.
Full solution
Q. Given the function , use the characteristics of polynomials and rational functions to describe its behavior and sketch the function.Enter the exact answers.Enter the intercepts as points, . Enter the -intercepts in increasing order of the -coordinate.The -intercepts are and The -intercept is The fields below accept a list of numbers or formulas separated by semicolons (e.g. or . The order of the list does not matter.
- Identify x-intercepts: Identify the x-intercepts by setting the numerator equal to zero and solving for :
Solve (double root)
Solve - Identify y-intercept: Identify the y-intercept by setting in the function:f(0) = \frac{(0 + 4)^2 * (0 - 2)}{2 * (0 - 1)^2 * (0 - 3)}\(\newline = \frac{(16 * -2)}{(2 * 1 * -3)} = \frac{-32}{-6} = \frac{32}{6} = \frac{16}{3}\)
- Identify vertical asymptotes: Identify vertical asymptotes by setting the denominator equal to zero and solving for :Solve (double root)Solve
- Identify horizontal asymptotes: Identify horizontal asymptotes by comparing the degrees of the numerator and the denominator:Degree of numerator = (from )Degree of denominator = (from )Since the degrees are equal, the horizontal asymptote is .
