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Math  /  Algebra

QuestionChoose the end behavior of the graph of each polynomial function. \begin{tabular}{|l|l|} \hline (a) f(x)=4x5−4x4+4x3−3xf(x)=4 x^{5}-4 x^{4}+4 x^{3}-3 x & Falls to the left and rises to the right \\ Rises to the left and falls to the right \\ Rises to the left and rises to the right \\ (b) f(x)=5x4−3x3−2x2−6f(x)=5 x^{4}-3 x^{3}-2 x^{2}-6 & Falls to the left and falls to the right and rises to the right \\ Rises to the left and falls to the right \\ (c) f(x)=−4(x−3)2(x+2)2f(x)=-4(x-3)^{2}(x+2)^{2} & Falls to the left and falls to the right and rises to the right \\ Rises to the left and falls to the right \\ Rises to the left and rises to the right rises to the right \\ Falls to the left and falls to the right \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The end behavior of a polynomial function is determined by the leading term.
2. The leading term is the term with the highest degree.
3. The sign of the leading coefficient and the degree of the polynomial determine the end behavior.

STEP 2

1. Identify the leading term of each polynomial.
2. Determine the degree of the polynomial.
3. Determine the sign of the leading coefficient.
4. Use the degree and sign to determine the end behavior.

STEP 3

Identify the leading term of each polynomial.
(a) For f(x)=4x5−4x4+4x3−3x f(x) = 4x^5 - 4x^4 + 4x^3 - 3x , the leading term is 4x5 4x^5 .
(b) For f(x)=5x4−3x3−2x2−6 f(x) = 5x^4 - 3x^3 - 2x^2 - 6 , the leading term is 5x4 5x^4 .
(c) For f(x)=−4(x−3)2(x+2)2 f(x) = -4(x-3)^2(x+2)^2 , expand to find the leading term. The leading term is −4x4 -4x^4 .

STEP 4

Determine the degree of each polynomial.
(a) The degree of f(x)=4x5−4x4+4x3−3x f(x) = 4x^5 - 4x^4 + 4x^3 - 3x is 5.
(b) The degree of f(x)=5x4−3x3−2x2−6 f(x) = 5x^4 - 3x^3 - 2x^2 - 6 is 4.
(c) The degree of f(x)=−4(x−3)2(x+2)2 f(x) = -4(x-3)^2(x+2)^2 is 4.

STEP 5

Determine the sign of the leading coefficient.
(a) The leading coefficient of 4x5 4x^5 is positive.
(b) The leading coefficient of 5x4 5x^4 is positive.
(c) The leading coefficient of −4x4 -4x^4 is negative.

STEP 6

Use the degree and sign to determine the end behavior.
(a) For f(x)=4x5−4x4+4x3−3x f(x) = 4x^5 - 4x^4 + 4x^3 - 3x , the degree is odd (5) and the leading coefficient is positive. Therefore, the end behavior is: Falls to the left and rises to the right.
(b) For f(x)=5x4−3x3−2x2−6 f(x) = 5x^4 - 3x^3 - 2x^2 - 6 , the degree is even (4) and the leading coefficient is positive. Therefore, the end behavior is: Rises to the left and rises to the right.
(c) For f(x)=−4(x−3)2(x+2)2 f(x) = -4(x-3)^2(x+2)^2 , the degree is even (4) and the leading coefficient is negative. Therefore, the end behavior is: Falls to the left and falls to the right.
The end behavior for each polynomial is: (a) Falls to the left and rises to the right. (b) Rises to the left and rises to the right. (c) Falls to the left and falls to the right.

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