Part 1Graph the polynomial function f(x)equals=negative 4 left parenthesis x plus 2 right parenthesis left parenthesis x minus 2 right parenthesis cubed−4(x+2)(x−2)3 using parts (a) through (e). Part 1(a) Determine the end behavior of the graph of the function.The graph of f behaves like yequals=[input]enter your response here for large values of StartAbsoluteValue x EndAbsoluteValuex.

Question

Part 1Graph the polynomial function ​f(x)equals=negative 4 left parenthesis x plus 2 right parenthesis left parenthesis x minus 2 right parenthesis cubed−4(x+2)(x−2)3 using parts​ (a) through​ (e). Part 1​(a) Determine the end behavior of the graph of the function.The graph of f behaves like yequals=[input]enter your response here for large values of StartAbsoluteValue x EndAbsoluteValuex.

BlackTom AI · Accepted Answer

The task presents a polynomial f(x) = -4(x+2)(x-2)^3 and asks for the end behavior, specifically a description like f(x) behaves like [leading term] for large |x|.

First, identify the leading term by considering the highest-degree terms from the factors. The factor (x+2) contributes a leading x, and (x-2)^3 contributes x^3 as its leading term when expanded. Multiplying these leading terms gives x * x^3 = x^4. The constant multiplier -4 scales this leading term, so the dominant term as |x| becomes large is -4x^4.

Because the polynomial has even degree (4) and a negative leading coefficient (-4), the end behavior is the same on both sides of the graph: as x → ∞ and as x → −∞, f(x) → −∞. In other words, the graph falls off toward negative infinity on both ends.

Additional context: when a polynomial is written in factored form, the end behavior is governed by the leading term obtained by multiplying the highest-degree parts of each factor. Here, you would expect the end-behavior approximation to be f(x) ~ -4x^4 for large |x|. This aligns with the standard rule that a polynomial with even degree and negative leading coefficient opens downward at both ends.

If you were to expand the expression, you would confirm the same leading term -4x^4, which reinforces the end-behavior conclusion. The exact function values for finite x would deviate from this leading-term approximation, but for the purpose of end behavior as |x| becomes large, the approximation accurately describes the sign and growth direction of f(x).

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