Question

Determine the equation of a rational function of the form \(y=\frac{a x+b}{c x+d}\) that has a vertical asymptote at \(x=6,\) a horizontal asymptote at \(y=-4,\) and an \(x\) -intercept of -1

Short answer

The equation of the rational function is \(y = \frac{-4x - 4}{x - 6}\).

Step-by-step solution

Identify the form of the rational function

The form of the rational function is given as \(y = \frac{ax + b}{cx + d}\).

Use the condition for the vertical asymptote

A vertical asymptote occurs where the denominator is zero. Since the vertical asymptote is at \(x = 6\), we have \(cx + d = 0\) when \(x = 6\). This implies \(6c + d = 0\). This gives us the equation \(d = -6c\).

Use the condition for the horizontal asymptote

The horizontal asymptote is found by considering the limits as \(x\) approaches infinity. For the given horizontal asymptote \(y = -4\), we set the leading coefficients of the numerator and denominator equal to the horizontal asymptote. This gives \(\frac{a}{c} = -4\). Therefore, \(a = -4c\).

Use the condition for the x-intercept

The x-intercept is found where the numerator is zero. Since the x-intercept is at \(x = -1\), we set \(ax + b = 0\) when \(x = -1\). Substituting \(a = -4c\), we have \(-4c(-1) + b = 0\) which simplifies to \(4c + b = 0\). Therefore, \(b = -4c\).

Write the final equation of the rational function

Substitute \(a = -4c\), \(b = -4c\), and \(d = -6c\) into the rational function form \(y = \frac{ax + b}{cx + d}\). We obtain \(y = \frac{-4cx - 4c}{cx - 6c}\). Factoring out \(c\) from numerator and denominator we get: \(y = \frac{c(-4x - 4)}{c(x - 6)}\) which simplifies to \(y = \frac{-4x - 4}{x - 6}\).

Key concepts

Vertical Asymptote

A vertical asymptote in a rational function occurs where the denominator is equal to zero, which means the function heads towards infinity as it approaches a specific x-value. For the given function, \(y = \frac{ax + b}{cx + d},\) a vertical asymptote is found when \(cx + d = 0.\) In the exercise, the vertical asymptote is located at \(x = 6.\) This implies that:\[cx + d = 0 \Rightarrow c(6) + d = 0 \Rightarrow 6c + d = 0,\]So, solving for \(d,\) we get:\[d = -6c.\] This indicates that every value of \(d\) related to the vertical asymptote can be expressed in terms of \(c.\)

Horizontal Asymptote

Horizontal asymptotes of a rational function are about understanding the behavior of the function as \(x\) approaches infinity or negative infinity. For the rational function \(y = \frac{ax + b}{cx + d},\) the horizontal asymptote is determined by the leading coefficients of the numerator and denominator. If the degrees of the numerator and denominator are the same, as in this case, the horizontal asymptote is found by dividing the leading coefficients. Given the horizontal asymptote of \(y = -4,\) we have:\[\frac{a}{c} = -4,\]which leads to: \[a = -4c.\]Thus, the coefficient \(a\) can be expressed in terms of \(c,\) ensuring the function aligns with the given horizontal asymptote.

x-intercept

The x-intercept of a rational function is where the function crosses the x-axis, which means where \(y = 0.\) For \(y = \frac{ax + b}{cx + d},\) this happens when the numerator is zero. In this exercise, the x-intercept is given as \(-1\). Hence, setting the numerator equal to zero when \(x = -1,\) we get: \[ax + b = 0 \text{ at } x = -1 \Rightarrow a(-1) + b = 0. \Rightarrow -a + b = 0.\]Using the relationship \(a = -4c,\) we substitute and solve: \[-4c + b = 0 \Rightarrow b = 4c.\]It shows how \(b\) is determined in terms of \(c,\) aligning the function with the given x-intercept.