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Horizontal asymptotes give us an idea of how a function behaves as the input value, represented by the variable usually denoted as \( x \), approaches positive infinity \( (+\infty) \) or negative infinity \( (-\infty) \). In the context of rational functions such as \( s(x) = \frac{8x^2+1}{4x^2+2x-6} \), the horizontal asymptote is determined by comparing the degrees of the numerator and the denominator.

For the function \( s(x) \), both the numerator \( 8x^2 + 1 \) and the denominator \( 4x^2 + 2x - 6 \) are quadratic, meaning they both have a degree of 2. In such cases, the horizontal asymptote is identified by dividing the leading coefficients of the numerator and the denominator.

• The leading coefficient of the numerator is 8.
• The leading coefficient of the denominator is 4.

The horizontal asymptote is thus \( y = \frac{8}{4} = 2 \).
This indicates that as \( x \) becomes very large or very small, the output \( s(x) \) approaches the line \( y = 2 \).

Vertical asymptotes are locations on the graph where the function approaches infinity or negative infinity, causing a break in the graph. They occur at roots of the denominator where the numerator is non-zero, making the expression undefined due to division by zero.

In the function \( s(x) = \frac{8x^2+1}{4x^2+2x-6} \), to find vertical asymptotes, solve the equation \( 4x^2 + 2x - 6 = 0 \).

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), where \( a = 4 \), \( b = 2 \), \( c = -6 \), you first compute the discriminant:
\[ b^2 - 4ac = 2^2 - 4(4)(-6) = 4 + 96 = 100 \]
With a positive discriminant, the function has two real roots, thus two vertical asymptotes.

Solving gives:
\[ x = \frac{-2 \pm \sqrt{100}}{8} \]
This simplifies to \( x = 1 \) and \( x = -\frac{3}{2} \).
These values represent the vertical asymptotes of the function. The graph of \( s(x) \) shoots to infinity or negative infinity as \( x \) approaches these values.

A quadratic function is a type of polynomial where the highest power of the variable \( x \) is 2. Such functions take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

The function \( s(x) = \frac{8x^2+1}{4x^2+2x-6} \) is built on two quadratic expressions, one serving as the numerator and the other as the denominator of a rational function.

Key characteristics of quadratic functions include:

• The graph of a quadratic function is a parabola.
• The direction of the parabola (opening upwards or downwards) depends on the sign of the coefficient \( a \). Positive \( a \) results in a parabola opening upwards, while negative \( a \) results in one that opens downwards.
• The roots of the quadratic function can be found using methods like factoring, completing the square, or using the quadratic formula.

Parabolas intersect the axes; the axis of symmetry and vertex can be essential points of interest. In rational functions, quadratic expressions in the denominator impact where the function might have vertical asymptotes, as solutions to the denominator equation \( 4x^2+2x-6=0 \) represent these asymptotic values.