Math Problem Statement

Find the rational function whose graph is shown in the figure below.

Solution

The graph shown suggests we need to find the rational function by analyzing the following key features:

  1. Vertical asymptotes: These occur where the denominator of the rational function equals zero. Based on the graph, we can see vertical asymptotes at x=2x = -2 and x=4x = 4. This suggests factors in the denominator of the form (x+2)(x + 2) and (x4)(x - 4).

  2. Horizontal asymptote: There seems to be a horizontal asymptote at y=0y = 0, which suggests the degree of the numerator is less than or equal to the degree of the denominator.

  3. Behavior near vertical asymptotes:

    • At x=2x = -2, the graph approaches positive infinity on the left and negative infinity on the right.
    • At x=4x = 4, the graph approaches negative infinity on the left and positive infinity on the right.
  4. Zeros of the function: The graph crosses the x-axis at x=2x = 2, suggesting a zero at x=2x = 2. This means the numerator contains a factor (x2)(x - 2).

Possible Rational Function

From this, we can hypothesize that the function is of the form:

f(x)=a(x2)(x+2)(x4)f(x) = \frac{a(x - 2)}{(x + 2)(x - 4)}

Determining the constant aa:

  • We also observe a point passing through (x=6,y=1)(x = 6, y = -1) on the graph. Substituting x=6x = 6 and y=1y = -1 into the function will allow us to solve for aa.

Let's calculate the value of aa.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes
Zeros of Functions

Formulas

Rational function: f(x) = a(x - 2) / ((x + 2)(x - 4))

Theorems

Vertical Asymptote Theorem
Horizontal Asymptote Theorem
Zero of a Function Theorem

Suitable Grade Level

Grades 10-12