What does the graph of the function f(x) = (-x^2 – 9x – 20) / (x + 4) look like?

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The function f(x) = \frac{-x^2 – 9x – 20}{x + 4} is a rational function, meaning it is the ratio of a polynomial in the numerator to a polynomial in the denominator. To understand the graph of this function, we’ll analyze several key features: the x-intercepts, the y-intercept, vertical and horizontal asymptotes, and the overall shape of the graph.

Step 1: Finding the x-intercepts

The x-intercepts of a function occur when f(x) = 0. For our function, this happens when the numerator is zero (since the only time a fraction is zero is when its numerator is zero, provided the denominator is not zero).
Therefore, we set the numerator equal to zero:

-x^2 - 9x - 20 = 0

To solve for x, we can factor this quadratic equation:

-(x^2 + 9x + 20) = 0 
=> x^2 + 9x + 20 = 0 
=> (x + 4)(x + 5) = 0

This gives us the solutions:
x = -4 and x = -5. Therefore, the x-intercepts of the function are the points (-4, 0) and (-5, 0).

Step 2: Finding the y-intercept

The y-intercept occurs when x = 0. Plugging this value into the function:

f(0) = \frac{-0^2 - 9(0) - 20}{0 + 4} = \frac{-20}{4} = -5

This means the y-intercept is at the point (0, -5).

Step 3: Identifying vertical asymptotes

Vertical asymptotes occur where the denominator is zero (provided the numerator is not also zero at that point). We examine:

x + 4 = 0 
=> x = -4

The function has a vertical asymptote at x = -4.

Step 4: Identifying horizontal asymptotes

To find horizontal asymptotes, we look at the degrees of the numerator and denominator. The numerator is a quadratic (degree 2) while the denominator is linear (degree 1). Since the degree of the numerator is greater than that of the denominator by 1, this tells us that:

  • There is no horizontal asymptote, but there is an oblique (slant) asymptote.

To find the equation of the oblique asymptote, we perform polynomial long division on the numerator divided by the denominator:

-x^2 - 9x - 20 divided by x + 4

This results in y = -x – 13.

Step 5: Sketching the Graph

Now we can summarize our findings:

  • X-intercepts: (-4, 0) and (-5, 0)
  • Y-intercept: (0, -5)
  • Vertical asymptote at x = -4
  • Oblique asymptote: y = -x – 13

To draw the graph:
– Start by plotting the intercepts and the vertical asymptote at x = -4.
– Notice the behavior of the graph as it approaches the vertical asymptote: f(x) will decrease without bounds as x approaches -4 from the left, and increase without bounds as it approaches from the right.
– The graph will follow the oblique asymptote as x moves toward positive and negative infinity.

In conclusion, the graph of the function is characterized by these features, creating a curve that wraps around the asymptotes and intercepts, forming a unique and engaging shape.

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