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Advanced Math / Nonlinear functions Difficulty: Medium

Which of the following could be the equation of the graph shown in the xy-plane?

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Explanation

Choice B is correct. Each of the given choices is an equation of the form $y = - \frac{1}{10} x {(x - a)}^{m} {(x + b)}^{n}$ , where $a$ , $b$ , $m$ , and $n$ are positive constants. In the xy-plane, the graph of an equation of this form has x-intercepts at $x equals 0$ , $x = a$ , and $x = - b$ . The graph shown has x-intercepts at $x equals 0$ , $x equals 4$ , and $x = - 5$ . Therefore, $a equals 4$ and $b equals 5$ . Of the given choices, only choices A and B have $a equals 4$ and $b equals 5$ . For an equation in the form $y = - \frac{1}{10} x {(x - a)}^{m} {(x + b)}^{n}$ , if all values of $x$ that are less than $- b$ or greater than $a$ correspond to negative y-values, then the sum of all the exponents of the factors on the right-hand side of the equation is even. In the graph shown, all values of $x$ less than $negative 5$ or greater than $4$ correspond to negative y-values. Therefore, the sum of all the exponents of the factors on the right-hand side of the equation $y = - \frac{1}{10} x {(x - 4)}^{m} {(x + 5)}^{n}$ must be even. For choice A, the sum of these exponents is $1 + 1 + 1$ , or $3$ , which is odd. For choice B, the sum of these exponents is $1 + 1 + 2$ , or $4$ , which is even. Therefore, $y = - \frac{1}{10} x (x - 4) {(x + 5)}^{2}$ could be the equation of the graph shown.

Choice A is incorrect. For the graph of this equation, all values of $x$ less than $negative 5$ correspond to positive, not negative, y-values.

Choice C is incorrect. The graph of this equation has x-intercepts at $x = - 4$ , $x equals 0$ , and $x equals 5$ , rather than x-intercepts at $x = - 5$ , $x equals 0$ , and $x equals 4$ .

Choice D is incorrect. The graph of this equation has x-intercepts at $x = - 4$ , $x equals 0$ , and $x equals 5$ , rather than x-intercepts at $x = - 5$ , $x equals 0$ , and $x equals 4$ .