The graph of f(x)=ax^2+bx+c has two distinct zeros. Explain how to sketch the graph of y=1/f(x).

Question

12th grade advanced functions math

Answer 1

The zeros of a graph will be vertical asymptotes of its reciprocal. Positive values stay positive, and negatives stay negative. Endpoints that go toward infinity will go toward zero, and vice versa.

 

As an example, g(x) = x² - 1 has zeros (-1, 0) and (1, 0). g(x) is positive on (-∞, -1) and (1, ∞) and negative on (-1, 1). The endpoints of g(x) both go toward +∞. The y-intercept is (0, -1)

 

The graph of h(x) = 1/g(x) then has the following characteristics:

-vertical asymptotes at x = -1 and x = 1

-a horizontal asymptote at y=0, both ends approaching from the positive side

-at x= -1, it will approach +∞ from the left, and -∞ from the right

-at x= 1, it will approach -∞ from the left, and +∞ from the right

-there will be no zeros (g(x) has no vertical asymptotes)

-the y-intercept is (0, 1/-1) = (0, -1)

 

Try sketching these out to make sure you understand the relationships. If you have further questions, please comment.