interpreting a function

I do not know how to do this question
Consider the function \(\displaystyle f(x) = [(2x-a)(x-b)]/[5(x-b)(x-c)]\) . Determine where the following features of its graph are located in terms of a, b, and/or c. Explain your reasoning for each situation.

a. the x-intercepts

b. the y-intercepts

c. any vertical asymptotes

d. any horizontal asymptotes

e. any “holes” in the graph

How do I find a b c d and e? does graphing it out make it easier? or are there formulas to use as well?
 

dwsmith

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I do not know how to do this question
Consider the function \(\displaystyle f(x) = [(2x-a)(x-b)]/[5(x-b)(x-c)]\) . Determine where the following features of its graph are located in terms of a, b, and/or c. Explain your reasoning for each situation.

a. the x-intercepts

b. the y-intercepts

c. any vertical asymptotes

d. any horizontal asymptotes

e. any “holes” in the graph

How do I find a b c d and e? does graphing it out make it easier? or are there formulas to use as well?
a. For x intercepts, we need to find points in the form (x,0) so lets do that.
\(\displaystyle f(x)=\frac{(2x-a)(x-b)}{5(x-b)(x-c)}=0\rightarrow (2x-a)(x-b)=2x^2-x(2b+a)+ab\)\(\displaystyle =(x-b)(2x-a)=0\)
You can finish it.

b. For y intercepts, we need to find points in the form (0,y).
\(\displaystyle f(0)=\frac{(2*0-a)(0-b)}{5(0-b)(0-c)}=\frac{ab}{5bc}\)

c. Vertical asymptotes will happen the fraction is undefined \(\displaystyle \frac{x}{0}\)
\(\displaystyle 5(x-b)(x-c)=0\rightarrow x=b,c\)

d. For this, we need to run the limits.
\(\displaystyle \lim_{x\to\infty}\frac{(2x-a)(x-b)}{5(x-b)(x-c)}\rightarrow \frac{2x^2}{5x^2}=\frac{2}{5} \)

e. This is actually giving to you since we are giving the fractions factored. This occurs when \(\displaystyle x=b\)
 
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