(x-2)(x-4)(x-7)/(x+2)(x+4)(x+7) 1

Jasonbradley

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Jun 17, 2024

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Analyzing the Expression: (x-2)(x-4)(x-7) / (x+2)(x+4)(x+7)

This expression represents a rational function, meaning a function that is defined as the ratio of two polynomials. Let's break down its key features and understand what it tells us.

Identifying the Roots and Vertical Asymptotes

• Roots: The roots of the function are the values of x that make the numerator equal to zero. In this case, the roots are x = 2, x = 4, and x = 7. These are the points where the graph of the function crosses the x-axis.

• Vertical Asymptotes: Vertical asymptotes occur where the denominator of the rational function equals zero. Here, the vertical asymptotes are located at x = -2, x = -4, and x = -7. These are the lines where the function approaches infinity as x gets closer to these values.

Analyzing the Behavior of the Function

• End Behavior: As x approaches positive or negative infinity, the function approaches 1. This is because the degree of the numerator and the denominator are the same (both are 3), and the leading coefficients are the same (1 in both cases).

• Sign Changes: The function will change sign at its roots and vertical asymptotes. This means that the graph will go from positive to negative or vice versa at these points.

Graphing the Function

To graph the function, you can use the information we've gathered.

• Plot the roots and vertical asymptotes.
• Consider the end behavior.
• Use the sign changes to determine where the graph is above or below the x-axis.

Remember, rational functions can be complex, but understanding the roots, vertical asymptotes, and end behavior provides a strong foundation for analyzing and graphing them.