To graph the rational function \(y = \frac{(x-4)(x-1)}{(x+5)(x-5)}\), we will first analyze the behavior of the function at certain critical points.
The function will have vertical asymptotes at \(x = -5\) and \(x = 5\) where the denominator becomes zero.
The function will have a hole at \(x = 4\) due to the factor \((x-4)\) in the numerator and denominator. The function becomes undefined at this point, but the hole can be filled in by cancelling out the factor.
The function will also have x-intercepts at \(x = 4\) and \(x = 1\) where the numerator becomes zero.
Now, plotting the critical points and knowing the behavior of the function near those points, we can sketch the graph. It will have a vertical asymptote at \(x = -5\) and \(x = 5\), a hole at \(x = 4\), and x-intercepts at \(x = 4\) and \(x = 1\). The graph will approach the asymptotes as \(x\) tends towards positive or negative infinity.
Overall, the graph of the rational function \(y = \frac{(x-4)(x-1)}{(x+5)(x-5)}\) will have a shape similar to a rational function with the above critical points taken into consideration.
What is the graph of the rational function?
y equals Start Fraction left parenthesis x minus 4 right parenthesis left parenthesis x minus 1 right parenthesis over left parenthesis x plus 5 right parenthesis left parenthesis x minus 5 right parenthesis End Fraction
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