Question

select the correct answer.
which statement describes the graph of the function $h(x)=\frac{4x^2 - 100}{8x - 20}$?
a. there is a horizontal asymptote at $y = 5$.
b. the graph has an oblique asymptote.
c. there is a horizontal asymptote at $y = 1$.
d. there is a horizontal asymptote at $y = \frac{1}{2}$.

Explanation:

Step1: Factor numerator and denominator

Factor numerator: $4x^2 - 100 = 4(x^2 - 25) = 4(x-5)(x+5)$
Factor denominator: $8x - 20 = 4(2x - 5)$
Simplify function: $h(x)=\frac{4(x-5)(x+5)}{4(2x-5)}=\frac{(x-5)(x+5)}{2x-5}$ (for $x
eq5$)

Step2: Check degrees of polynomials

Degree of numerator (2) > Degree of denominator (1)

Step3: Analyze asymptote type

When numerator degree > denominator degree by 1, the graph has an oblique (slant) asymptote, not a horizontal asymptote.

Answer:

B. The graph has an oblique asymptote.