Question

Find the slant asymptote of the graph of the rational function and b. Use the slant asymptote to graph the rational function. f(x)=x^2-x-20/x-7

Answer 1

The rational function is f(x) = (x² - x - 20)/(x - 7).

Step 1: Factor the numerator and denominator of f(x). Find the domain of the rational function.

write the numarator (x² - x - 20) in facotr from.

x² - x - 20 = x² - 5x + 4x - 20

                 = x(x - 5) + 4(x - 5)

                 = (x + 4)(x - 5)

f(x) = [(x + 4)(x - 5)]/(x - 7)

The domain of f(x) is { x | x ≠ 7 }.

Step 2: Write f(x) in lowest terms.

Because there are no common factors between the numerator and denominator, f(x) is in lowest terms.

Step 3: Locate the intercepts of the graph.

Since 0 is in the domain of f(x), the y-intercept is f(0) = 20/7. The x-intercepts are found by determining the real zeros of the numerator of f(x) that are in the domain of f(x).  By solving (x + 4)(x - 5) = 0, the only real zero of the numerator are - 4 and 5, so the only x-intercept of graph of f(x) are - 4 and 5.

We analyze the behavior of the graph of f(x)

Near - 4 :

f(x) = [(x + 4)(x - 5)]/(x - 7) ≈ [(x + 4)(- 4 - 5)]/(- 4 - 7) = (9/11)(x + 4)

Near 5 :

f(x) = [(x + 4)(x - 5)]/(x - 7) ≈ [(5 + 4)(x - 5)]/(5 - 7) = - (9/2)(x - 5).

Plot the point (- 4, 0) and indicate a line with positive slope there. Plot the point (5, 0) and indicate a line with negative slope there.

Step 4: Locate the vertical asymptotes.

To  find any vertical asymptotes, set the denominator equal to zero and solve the resulting equation for x.

The denominator = x - 7 = 0 ⇒ x = 7. The line x = 7 is the vertical asymptote of the graph of f(x).

Step 5: Locate the horizontal or oblique asymptote, if one exists.

Since the degree of the numerator, 2, is greater than the degree of the denominator, 1, the rational function f(x) is improper. To find a horizontal or oblique( slant ) asymptote, we use long division.

The line y = x + 6 is the horizontal or oblique asymptote of the graph of f(x).

To determine whether the graph of R intersects the asymptote y = x + 6, we solve the equation f(x) = x + 6

f(x) = [(x + 4)(x - 5)]/(x - 7) = x + 6

(x + 4)(x - 5) = (x + 6)(x - 7).

x² - x - 20 = x² - x - 42

20 = 42

This is not true, we conclude that the equation [(x + 4)(x - 5)]/(x - 7) = x + 6 has no solution, so  the graph of R does not intersect the line y = x + 6.

Step 6: Use the zeros of the numerator and denominator of f(x) to divide the x-axis into intervals.

The zeros of the numerator, - 4 and 5 ; and the zero of the denominator is 7 , divide the x-axis into four intervals:

(- ∞, - 4), (- 4, 5), (5, 7) and (7, ∞)

Now construct Table :

Interval x-value   f(x) = [(x + 4)(x - 5)]/(x - 7)      location of graph  Point on the graph

(-∞, -4)  x = -15   f(-15) = [(-15 + 4)(-15 - 5)]/(-15 - 7) =  -10 < 0  Below x-axis             (-15, -10)

(- 4, 5)     x = 2     f(2) = [(2 + 4)(2 - 5)]/(2 - 7) = 18/5 > 0               Above x-axis              (0, 20/7)

(5, 7)       x = 6     f(6) = [(6 + 4)(6 - 5)]/(6 - 7) = - 10 < 0                Below x-axis              (6, 10)

(7, ∞)      x = 8     f(10) = [(10 + 4)(10 - 5)]/(10 - 7) = 70/3 > 0      Above x-axis                (8, 36)

Step 7: Analyze the behavior of the graph of f(x) near each asymptote and indicate this behavior on the graph.

Since the graph of f(x) is below the x-axis for x <  - 4 and is above the x-axis for x > 7, and since the graph of f(x) does not intersect the oblique asymptote , the graph of f(x) will approach the line as shown in the figure.

Since the graph of f(x) is above the x-axis for - 4 < x < 5, the graph of f(x) will approaches to vertical asymptote x = 7; since the graph of f(x) is below the x-axis for 6 < x < 10, the graph of f(x) will approaches to vertical asymptote x = 7.

Step 8 : Graph