Based only on the vertical asymptotes, which of the following graphs could be the graph of the given function? f(x)=x2−x−12x2−4x
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Identify the vertical asymptotes of the function \( f(x) = \frac{x^2 - 4x}{x^2 - x - 12} \). Vertical asymptotes occur where the denominator is zero and the numerator is not zero.
Factor the denominator \( x^2 - x - 12 \) to find the values of \( x \) that make it zero. The expression factors as \( (x - 4)(x + 3) \).
Set each factor equal to zero to find the vertical asymptotes: \( x - 4 = 0 \) gives \( x = 4 \) and \( x + 3 = 0 \) gives \( x = -3 \).
Check the numerator \( x^2 - 4x \) at these points to ensure they are not zero. Substitute \( x = 4 \) and \( x = -3 \) into the numerator to verify they do not equal zero, confirming vertical asymptotes at these points.
Compare the vertical asymptotes \( x = 4 \) and \( x = -3 \) with the graphs provided. The correct graph will have vertical lines (asymptotes) at these x-values.