Find the y- intercept f(x)=x^2-x-20/x-7

Question

Find the x-intercept(s),

Find the vertical asymptote(s) x=

Find the horizontal asymptote(s) y=

Answer 1

(1).

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

To find the x-intercept, the function f(x) = 0.

(x² - x - 20)/(x - 7) = 0

x² - x - 20 = 0

x² - 5x + 4x - 20 = 0

x(x - 5) + 4(x - 5) = 0

(x + 4)(x - 5) = 0

x + 4 = 0 and x - 5 = 0

x = - 4 and x = 5

The x-intercepts are - 4 and 5.

 

(2).

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).

Answer 2

(3).

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).