Nonlinear functions SAT question

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 $x=0$, $x=a$, and $x=-b$. The graph shown has x-intercepts at $x=0$, $x=4$, and $x=-5$. Therefore, $a=4$ and $b=5$. Of the given choices, only choices A and B have $a=4$ and $b=5$. 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 $-5$ 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 $-5$ correspond to positive, not negative, y-values.

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

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