Question

use the a, b, and c - sliders to graph the function ( f(x)=\frac{x - 1}{x^{2}-4x + 3}(a=-1,b=-4,\text{ and }c = 3) ). note that the denominator factors to ( (x - 3)(x - 1) ). complete parts 1 and 2 below.
use the interactive figure to find your answer. use the left and right arrow keys to move along a slider as needed.
click here to launch the interactive figure.
part 1: what is the equation of the vertical asymptote?
a. ( y = 3x )
b. ( x = 3 )
c. ( x = y )
d. ( y = 0 )

Explanation:

Step1: Recall the definition of vertical asymptote

A vertical asymptote occurs at the value of \(x\) that makes the denominator of a rational function equal to zero (provided the numerator is not also zero at that value).

Step2: Set the denominator equal to zero

The function is \(f(x)=\frac{x - 1}{x^{2}-4x + 3}\), and the denominator \(x^{2}-4x + 3=(x - 3)(x - 1)\). Set \(x^{2}-4x + 3 = 0\), so \((x - 3)(x - 1)=0\).

Step3: Solve for \(x\)

Using the zero - product property \(ab = 0\) implies \(a=0\) or \(b = 0\). So \(x-3=0\) gives \(x = 3\) and \(x - 1=0\) gives \(x = 1\). But when \(x = 1\), the numerator \(x-1=0\). So the vertical asymptote is at \(x = 3\) (since the numerator is non - zero at \(x = 3\)).

Answer:

B. \(x = 3\)