Correct Answer
Option B (y equals the opposite of x squared open parenthesis x plus 5 close parenthesis open parenthesis x minus 2) \( \left(y = -x^2\,(x+5)(x-2)\right) \) should be selected.
Explanation
The graph present a fourth-order polynomial function. We know it is a fourth-order polynomial because the graph changes direction three times.
We can eliminate C and D by looking at the factors. For example, if open parenthesis x plus 2 close parenthesis \( (x+2) \) were a factor of f of x \( f(x), \) then we would need x plus 2 equals 0, so x equals negative 2. \( x+2 = 0, \text{ so } x = -2. \) The graph shows no root at x equals negative 2, \( x = -2, \) so C and D can be ruled out.
The choice between A and B comes down to the behavior of the function as x gets larger and larger. Does it go up toward positive infinity \( +\infty \) or down toward negative infinity? \( -\infty? \) As x increases (on the right side of the graph), the leading term (the term with the highest power of x) is the dominant term. If the sign of the coefficient for the leading term is negative, the graph will go down as x increases; if the sign is positive, the graph will go up as x increases.
This graph shows that as x gets bigger and bigger, the function gets more and more negative. That means the coefficient on the leading term is negative.
Only options B and D have a negative coefficient on the leading term, as you can learn by expanding them. Since option D has been eliminated because the function has no root at 5 or negative 2, \( -2, \) only option B is left.
