Math  /  Algebra

QuestionChoose the end behavior of the graph of each polynomial function.

Studdy Solution

STEP 1

1. The end behavior of a polynomial function is determined by the leading term.
2. The degree and the leading coefficient of the polynomial function affect its end behavior.

STEP 2

1. Determine the degree and leading coefficient of each polynomial.
2. Use the degree and leading coefficient to determine the end behavior.

STEP 3

For polynomial (a) f(x)=5x35x29x3 f(x) = 5x^3 - 5x^2 - 9x - 3 , identify the leading term.
The leading term is 5x3 5x^3 .

STEP 4

For polynomial (a), the degree is 3 (odd) and the leading coefficient is positive.
Since the degree is odd and the leading coefficient is positive, the end behavior is: - Falls to the left and rises to the right.

STEP 5

For polynomial (b) f(x)=2x67x5+3x+8 f(x) = -2x^6 - 7x^5 + 3x + 8 , identify the leading term.
The leading term is 2x6 -2x^6 .

STEP 6

For polynomial (b), the degree is 6 (even) and the leading coefficient is negative.
Since the degree is even and the leading coefficient is negative, the end behavior is: - Falls to the left and falls to the right.

STEP 7

For polynomial (c) f(x)=3(x1)(x+5)2 f(x) = -3(x-1)(x+5)^2 , expand to find the leading term.
The expanded form has a leading term of 3x3 -3x^3 .

STEP 8

For polynomial (c), the degree is 3 (odd) and the leading coefficient is negative.
Since the degree is odd and the leading coefficient is negative, the end behavior is: - Rises to the left and falls to the right.
The end behaviors are: (a) Falls to the left and rises to the right. (b) Falls to the left and falls to the right. (c) Rises to the left and falls to the right.

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