Intercepts and Asymptotes

Question

Find all of the intercepts and asymptotes of the graph of f(x)= 5x+4/x+2. Be sure to label what type what type pf intercept and asymptote each is

Answer 1

If you have to find asymptotes of the function, then the function is likely to be a rational function.  It should be read as

 

f(x) = (5x + 4) / (x + 2)

 

 

y-intercept is (0, 2)

 

To find the x-intercept, set the numerator part of f(x) equal to zero and solve for x.

 

5x + 4 = 0     

 

5x = -4

 

x = -4/5    ----->  A real solution

 

x-intercept is (-4/5 , 0)

 

To find the horizontal asymptote, we look at the degree of the numerator and denominator.  The degree in the numerator and denominator is 1.  Because the degrees are equal to each other, we simply divide the coefficients of the leading terms.

 

y = 5 / 1

 

y = 5

 

The horizontal asymptote is     y=5

 

 

In the future, use parentheses to make your mathematical expressions and equations more clearer.  If parentheses are not present, then the answers you are given will be inaccurate.  But the answer I am providing is the correct one.

 

As you can see, Carmen's interpretation of the function led her to the conclusion that there are no x-intercepts.  This is because of the ambiguity of the function.

Answer 2

Hi Lauren

 

To find the y intercept you have to make x=0, however you have a term 4/x that goes indeterminate when x=0 and therefore x is not in the domain of the function, x=0 is a vertical asymptote.  There is no y intercept. 

 

To find the x intercepts, you have to set y=f(x)=0,  then 5x+4/x+2=0  you can multiply the whole equation by x and that give you the quadratic equation 5x^2+2x+4.  This equation do not have real solutions therefore the graphs do not have x intercepts.  To verify solve the quadratic discriminant b^2-4ac, since is negative there are not real solutions.  There are no horizontal asymptotes. 

 

Hope this helps!

 

Carmen