Question

use the drawing tools to form the correct answers on the graph. draw the lines representing the vertical and horizontal asymptotes on the graph of this function. $f(x)=\frac{2x^4 + 7x - 4}{x^2 + 5x + 4}$

Explanation:

Step1: Factor numerator and denominator

Factor $2x^2+7x-4$: $2x^2+7x-4=(2x-1)(x+4)$
Factor $x^2+5x+4$: $x^2+5x+4=(x+1)(x+4)$
Simplify $f(x)$: $f(x)=\frac{(2x-1)(x+4)}{(x+1)(x+4)}=\frac{2x-1}{x+1}, \ x
eq-4$

Step2: Find vertical asymptote

Set denominator to 0: $x+1=0$
Solve for $x$: $x=-1$

Step3: Find horizontal asymptote

Compare degrees of numerator/denominator (both 1). Take ratio of leading coefficients: $\frac{2}{1}=2$
Horizontal asymptote: $y=2$

Answer:

Vertical asymptote: $x=-1$ (vertical line through $x=-1$)
Horizontal asymptote: $y=2$ (horizontal line through $y=2$)