To determine where the graph of the function [tex]\( f(x) = \frac{x^2 + 4x + 6}{4x^2 + 8x - 10} \)[/tex] intersects its horizontal asymptote, follow these steps:
### Step 1: Find the Horizontal Asymptote
For rational functions, the horizontal asymptote is found by examining the degrees of the polynomials in the numerator and the denominator:
- The degree of the numerator [tex]\( x^2 + 4x + 6 \)[/tex] is 2.
- The degree of the denominator [tex]\( 4x^2 + 8x - 10 \)[/tex] is also 2.
Since the degrees of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the leading coefficients.
The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 4. Therefore, the horizontal asymptote is:
[tex]\[
y = \frac{1}{4}
\][/tex]
### Step 2: Determine Whether the Graph Intersects the Horizontal Asymptote
We need to determine if there is an [tex]\( x \)[/tex]-value such that [tex]\( f(x) = \frac{1}{4} \)[/tex]. Set the function equal to the horizontal asymptote and solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{x^2 + 4x + 6}{4x^2 + 8x - 10} = \frac{1}{4}
\][/tex]
Cross-multiply to clear the fraction:
[tex]\[
4(x^2 + 4x + 6) = 1(4x^2 + 8x - 10)
\][/tex]
Distribute and simplify both sides:
[tex]\[
4x^2 + 16x + 24 = 4x^2 + 8x - 10
\][/tex]
Subtract [tex]\( 4x^2 \)[/tex] from both sides:
[tex]\[
16x + 24 = 8x - 10
\][/tex]
Subtract [tex]\( 8x \)[/tex] from both sides:
[tex]\[
8x + 24 = -10
\][/tex]
Subtract 24 from both sides:
[tex]\[
8x = -34
\][/tex]
Divide both sides by 8:
[tex]\[
x = -\frac{34}{8} = -\frac{17}{4}
\][/tex]
Thus, the graph of the function [tex]\( f(x) \)[/tex] intersects the horizontal asymptote [tex]\( y = \frac{1}{4} \)[/tex] at [tex]\( x = -\frac{17}{4} \)[/tex].
### Conclusion
The correct answer is:
a) [tex]\( x = -\frac{17}{4} \)[/tex]
