How do you graph #f(x)= (x^2 +10x+ 24)/( x+6)#?

Question

Avery Alexander · Accepted Answer

You can factorise the quadratic term and see where that gets you.

First we note the restriction that #x!=-6# #=((x+4)(x+6))/(x+6)# #=((x+4)cancel((x+6)))/cancel(x+6)=x+4#

Eli Assouline · Answer

undefined To graph the function f(x) = (x^2 + 10x + 24)/(x + 6), follow these steps:

1. Determine the vertical asymptote: Set the denominator (x + 6) equal to zero and solve for x. The vertical asymptote occurs at x = -6.

2. Determine the horizontal asymptote: Compare the degrees of the numerator and denominator. Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote.

3. Find the x-intercepts: Set the numerator (x^2 + 10x + 24) equal to zero and solve for x. The x-intercepts occur when x = -4 and x = -6.

4. Find the y-intercept: Substitute x = 0 into the function to find the y-intercept. The y-intercept is (0, 4).

5. Plot the points: Plot the vertical asymptote at x = -6, the x-intercepts at x = -4 and x = -6, and the y-intercept at (0, 4).

6. Determine the behavior of the graph: As x approaches negative infinity or positive infinity, the function approaches the vertical asymptote at x = -6.

7. Sketch the graph: Connect the plotted points and draw a smooth curve that approaches the vertical asymptote at x = -6.

This is the graph of f(x) = (x^2 + 10x + 24)/(x + 6).