Problem

Plot all of the existing five features of the following rational function (some may not be needed). If you get a fraction or decimal then plot as close to the true location as possible. \[ f(x)=\frac{-3 x-9}{x^{2}-2 x-15} \] Plot Rational Function Click on a feature then drag it into place.

Solution

Step 1 :Set the denominator equal to zero and solve for \(x\): \(x^2 - 2x - 15 = 0\)

Step 2 :Factor the equation: \((x - 5)(x + 3) = 0\)

Step 3 :Solve for \(x\) to find the vertical asymptotes: \(x = 5, -3\)

Step 4 :Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is \(y = 0\)

Step 5 :Set the numerator equal to zero and solve for \(x\) to find the x-intercepts: \(-3x - 9 = 0\), \(-3x = 9\), \(x = -3\)

Step 6 :Set \(x = 0\) in the function to find the y-intercept: \(f(0) = -3(0) - 9 / (0)^2 - 2(0) - 15 = -9 / -15 = 3/5\)

Step 7 :There are no x-values where both the numerator and the denominator of the rational function equal zero, so there are no holes in the graph

Step 8 :The graph of the function has vertical asymptotes at \(x = 5\) and \(x = -3\), a horizontal asymptote at \(y = 0\), an x-intercept at \(x = -3\), a y-intercept at \(y = 3/5\), and no holes. \(\boxed{\text{Final Answer}}\)

From Solvely APP
Source: https://solvelyapp.com/problems/9VL2sdpKWd/

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