Question

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Which equation could represent each graphed polynomial function?
(A) y = x ( x + 3 ) ( x - 2 )
(B) y = ( x + 1 ) ( x - 3 ) ( x^2 + 1 )
(C) y = x^3 + 27
(D) y = x^4 - 5x^2 + 4

Answer 1

To match a graphed polynomial function to its equation, analyze the roots and end behavior of the graph, then compare these to the characteristics suggested by the given equations. Cubic and quartic functions differ in the number of potential real roots and changes in concavity, which can help identify the correct match.

Step-by-step explanation:

The question involves understanding how polynomial functions are graphically represented by their equations. To match a graphed polynomial function with its corresponding equation, consider the number of roots (x-intercepts), end behavior, and the presence of complex roots (which would not cross the x-axis on a real graph). Looking at the provided options:

• Option A is a cubic function with roots at x=-3, x=0, and x=2.
• Option B is a quartic function that factors into a quadratic expression, (x²+1), which does not have real roots, and two linear factors suggesting two real roots.
• Option C is an expression of a cubic function with just one real root, which can be found by factoring y=x³+27 into y=(x+3)³.
• Option D represents a quartic function with a probable change in concavity indicated by the x´ term and potential real roots suggested by the constant term.

To match each graph with its respective equation, check the number of x-intercepts and the end behavior of each function in the graphs provided, then compare these characteristics with those suggested by the form of each of the given equations.