In quadrant 3: The curve rises sharply to touch the x axis at point (negative 5 comma 0). The curve

Question

In quadrant 3:
- The curve rises sharply to touch the x axis at point (negative 5 comma 0).
- The curve falls gradually to a relative minimum at point (negative 2 comma negative 11).
- The curve rises gradually to cross both axes at the origin.
In quadrant 1:
- The curve rises gradually to a relative maximum at point (2.5 comma 21).
- The curve falls sharply to cross the x axis at point (4 comma 0).
In quadrant 4 the curve falls sharply.

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

Accepted Answer

Choice B is correct. Each of the given choices is an equation of the form y = - 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 = - 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 = - 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 = - 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 .

Question 30c5bb