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Vertical asymptotes serve as invisible walls along which the graph of a rational function shoots up or down toward infinity. If our rational function has a vertical asymptote at \( x=5 \), this occurs because the denominator of the function equals zero when \( x=5 \). It's just like hitting something solid in math terms—an undefined point.

Therefore, the denominator must include a factor that zeros out when \( x=5 \). The simplest factor that achieves this is \( x-5 \). So, by simply stating "a vertical asymptote at \( x=5 \)," you know the denominator of your function must have \( x-5 \) as a factor. This makes it impossible for the function to be continuous at that point, causing the graph to shoot upward or downward as it approaches \( x=5 \).

Horizontal asymptotes give us insight into the behavior of a function as \( x \) tends towards positive or negative infinity. When you see a horizontal asymptote at \( y = -1 \), it tells you that as the values of \( x \) become extremely large or small, the function approaches \(-1\) as its value.

Mathematically, this occurs when the degrees of the numerator and denominator are equal. With equal degrees, the horizontal asymptote is determined by the ratio of their leading coefficients. Here, the equation \( y = -1 \) tells us that the leading coefficients of the numerator and denominator must have a ratio of \(-1:1\).

This means that for our particular rational function, if the numerator and denominator both have a degree of 1, the leading coefficient of the numerator must be \(-1\) to maintain a horizontal asymptote of \(-1\). Further, this directs us to construct the numerator as \(-1(x - c)\) for simplicity.

X-intercepts occur where the graph crosses the x-axis, meaning the output of the function (\( y \)) is zero at these points. For the given \( x \)-intercept at \( x=2 \), the function \( f(x) \) becomes zero when \( x=2 \).

To factor this into our rational function, we place \( x-2 \) in the numerator because setting the numerator equal to zero will make the function itself equal to zero, thus providing the x-intercept. If you have an x-intercept at a particular point, the term connected with that x-value in the numerator of the equation is simply \( (x - \, \text{[x-intercept]}) \).

Hence, the numerator should include \( x-2 \) so that when \( x \) equals 2, the numerator equals zero, and thus \( f(x) \) equals zero, achieving the intercept desired.

Constructing rational functions involves weaving together the pieces of a mathematical puzzle. For this problem, each condition gives us key parts of the function.

• The vertical asymptote at \( x = 5 \) means \( x - 5 \) is a factor of the denominator.
• The horizontal asymptote at \( y = -1 \) indicates the degrees of the numerator and denominator are the same, with the numerator's leading coefficient being \(-1\).
• The x-intercept at \( x = 2 \) demands \( x - 2 \) be a factor of the numerator.

Armed with this knowledge, we can articulate our rational function. This leads us to the elegant formula \( f(x) = \frac{-(x-2)}{x-5} \), which satisfies all the constraints. Ensuring each trait of the function (asymptotes and intercepts) are aligned with our requirements is the essence of constructing such equations. By analyzing and applying each condition, we bring clarity and coherence to the function's formulation.