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Vertical asymptotes are important because they tell us where the function is undefined and how the function behaves near these points. A vertical asymptote occurs when the denominator of a rational function is zero because division by zero is undefined. For the function given, \(f(x) = \frac{1}{(x+5)(x-2)}\), the denominator is \((x+5)(x-2)\). To find the vertical asymptotes, we solve \((x + 5)(x - 2) = 0 \). This gives us two points: \ x = -5 \ and \ x = 2 \.
These are the vertical asymptotes. When you graph the function, the curve will approach these lines but will never touch or cross them. As you get very close to these points from either side, the function will increase or decrease without bound. This behavior is crucial to understand the overall shape of the graph.

The horizontal asymptote of a rational function provides information about the end behavior of the function as \(x \) approaches positive or negative infinity. To find a horizontal asymptote, we compare the degrees of the numerator and the denominator. For the function \(f(x) = \frac{1}{(x+5)(x-2)}\), the degree of the numerator (which is 0) is less than the degree of the denominator (which is 2).
Hence, the horizontal asymptote is \ y = 0 \. This means that as \ x \ approaches infinity or negative infinity, the function \ f(x) \ will get closer and closer to zero but will never actually touch or cross the \ y = 0 \ line. Understanding where the horizontal asymptote is helps to sketch the end behavior of the graph accurately.

Intercepts are the points where the graph crosses the axes. There are two types: x-intercepts and y-intercepts. For the given function \(f(x) = \frac{1}{(x+5)(x-2)}\), the y-intercept can be found by setting \ x = 0 \ in the function.
So, \ f(0) = \frac{1}{(0+5)(0-2)} = \frac{1}{-10} = -\frac{1}{10} \). Therefore, the y-intercept is \( (0, -\frac{1}{10}) \.
The x-intercepts are found by setting the numerator equal to zero. Since the numerator is 1, which never equals zero, the function has no x-intercepts.
Identifying intercepts helps to anchor the graph and provide key points through which the graph will pass.

End behavior describes how the function acts as \ x \ approaches infinity or negative infinity. For the rational function \(f(x) = \frac{1}{(x+5)(x-2)}\), we look at how the forces of the numerator and denominator play out at extreme values of \ x \.
Since the denominator grows much faster than the constant numerator (1), the value of \( f(x) \) will get smaller and smaller, approaching the horizontal asymptote \ y = 0 \.
This means the function will approach zero as \ x \ goes to either positive or negative infinity. This tells us that the function will get very close to the x-axis but will not touch it. Knowing the end behavior assists in correctly plotting the graph over a broad set of values.