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3.3e: Exercises - Polynomial End Behaviour

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    45438
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    A: Concepts

    Exercise \(\PageIndex{A}\) 

    1) Explain the difference between the coefficient of a power function and its degree.

    2) If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

    3) In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

    4) What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As \(x \rightarrow-\infty, f(x) \rightarrow-\infty\) and as \(x \rightarrow \infty, f(x) \rightarrow-\infty\).

    Answers to odd exercises:

    1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

    3. As \(x\) decreases without bound, so does \(f(x)\). As \(x\) increases without bound, so does \(f(x)\).

    B: Identify Power Functions and Polynomials

    Exercise \(\PageIndex{B}\) 

    \( \bigstar \) Identify the function as a power function, a polynomial function, or neither. If neither, explain.

    5) \(f(x)=(x^2)^3\)

    6) \(f(x)=x^5\)

    7) \(f(x)=x−x^4\)

    8) \(f(x)=\dfrac{x^2}{x^2−1}\)

    9) \(f(x)=3^{x+1}\)

    10) \(f(x)=2x(x+2)(x−1)^2\)

    Answers to odd exercises:

    5. Power function, \(\quad\) 7. Polynomial, \(\quad\) 9. Neither - this is an exponential function because the variable is in the exponent

    C: Degree and Leading Coefficient of a Polynomial

    Exercise \(\PageIndex{C}\) 

    \( \bigstar \) (a) Find the degree and leading coefficient for the given polynomial. (b) State the end behaviour.

    11) \(7−2x^2\)

    12) \(−3x^4\)

    13) \(x(4−x^2)(2x+1)\)

    14) \(−2x^2− 3x^5+ x−6\)

    15) \(x^2(2x−3)^2\)

    16) \( -3(2x+5)^2(1-x)^3 \)

    Answers to odd exercises:
    11. Degree \(2\), Coefficient = \(-2\) \( \swarrow \dots \searrow \) 13. Degree \(4\), Coefficient = \(-2\) \( \swarrow \dots \searrow \) 15. Degree \(4\), Coefficient = \(4\)  or  \( \nwarrow \dots \nearrow\)

    D: Polynomial End Behaviour

    Exercise \(\PageIndex{D}\) 

    \( \bigstar \) Determine the end behavior of the functions.

    17) \(f(x)=x^4\)

    18) \(f(x)=x^3\)

    19) \(f(x)=−x^4\)

    20) \(f(x)=−x^9\)

     

    21) \(f(x)=−2x^4− 3x^2+ x−1\)

    22) \(f(x)=3x^2+ x−2\)

    23) \(f(x)=x^2(2x^3−x+1)\)

    24(a) \(f(x)=(2−x)^7\)

    24(b) \(f(x)=3x(2x-3)^4\)

    25) \(f(x)=x^4−5x^2\)

    26) \(f(x)=−x^3\)

    27) \(f(x)=(x−1)(x−2)(3−x)\)

    28) \(f(x)=x^2(1−x)^2\)

    29) \(f(x)=\dfrac{x^5}{10}−x^4\)

    Answers to odd exercises:

    17. As \(x→\pm ∞\), \(f(x)→∞\), \(\quad\)or  \( \nwarrow \dots \nearrow\)

    19. As \(x→\pm ∞\), \(f(x)→−∞\), \(\quad\)or  \( \swarrow \dots \searrow \)

    21. As \(x→\pm∞\), \(f(x)→−∞\), \(\quad\)or  \( \swarrow \dots \searrow \)

    23. As \(x→∞\), \(f(x)→∞\), as \(x→−∞\), \(f(x)→−∞  \)
    \(\quad \; \) or  \( \swarrow \dots \nearrow \)

    25. As \(x→\pm ∞\), \(f(x)→∞\), or  \( \nwarrow \dots \nearrow\)

    27. as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→−∞\)
    \(\quad \; \) or  \( \nwarrow \dots \searrow  \)

    29.  As \(x→∞\), \(f(x)→∞\), as \(x→−∞\), \(f(x)→−∞  \)
    \(\quad \; \) or  \( \swarrow \dots \nearrow \)

     

    E: Polynomial Intercepts

    Exercise \(\PageIndex{E}\) 

    \( \bigstar \) Find the intercepts of the functions.

    30) \(C(t)=2(t−4)(t+1)(t−6)\)

    31) \(f(t)=2(t−1)(t+2)(t−3)\)

    32) \(g(n)=−2(3n−1)(2n+1)\)

    33) \(f(x)=x^4−16\)

    34) \(f(x)=x^3+27\)

    35) \(f(x)=x(x^2−2x−8)\)

    36) \(f(x)=(x+3)(4x^2−1)\)

    37) \(C(t)=3(t+2)(t−3)(t+5)\)

    38) \(C(t)=4t(t−2)^2(t+1)\)

    39) \(C(t)=2t(t−3)(t+1)^2\)

    40) \(C(t)=2t^4−8t^3+6t^2\)

    41) \(C(t)=4t^4+12t^3−40t^2\)

    42) \(f(x)=x^4−x^2\)

    43) \(f(x)=x^3+x^2−20x\)

    44) \(f(x)=x^3+6x^2−7x\)

    45) \(f(x)=x^3+x^2−4x−4\)

    46) \(f(x)=x^3+2x^2−9x−18\)

    47) \(f(x)=2x^3−x^2−8x+4\)

    48) \(f(x)=x^6−7x^3−8\)

    49) \(f(x)=2x^4+6x^2−8\)

    50) \(f(x)=x^3−3x^2−x+3\)

    51) \(f(x)=x^6−2x^4−3x^2\)

    52) \(f(x)=x^6−3x^4−4x^2\)

    53) \(f(x)=x^5−5x^3+4x\)

    \( \bigstar \) Determine the intercepts and the end behavior.

    55) \(f(x)=x^3(x−2)\)

    56) \(f(x)=x(x−3)(x+3)\)

    57) \(f(x)=x(14−2x)(10−2x)\)

    58) \(f(x)=x(14−2x)(10−2x)^2\)

    59) \(f(x)=x^3−16x\)

    60) \(f(x)=x^3−27\)

    61) \(f(x)=x^4−81\)

    62) \(f(x)=−x^3+x^2+2x\)

    63) \(f(x)=x^3−2x^2−15x\)

    64) \(f(x)=x^3−0.01x\)

    Answers to odd exercises:

    31.\((0,12)\); \((1,0),\;(–2,0),\; (3,0)\)

    33. \((0,−16)\); \((2,0),\; (−2,0)\).

    35. \((0,0), \; (4,0), \; (−2, 0)\).

    37. \((0, 90)\); \((−2,0),\;(3,0),\;(−5,0)\)

    39. \((0,0)\), \((3,0),(−1,0) \)

    41. \((0,0), (−5,0),\; (2,0)\)

    43. \((0,0),\; (−5,0),\; (4,0)\)

    45. \((0, -4)\); \((2,0),\; (−2,0),\; (−1,0)\)

    47. \((0, 4)\);  \((−2,0),\;(2,0),\left(\tfrac{1}{2},0\right)\)

    49. \((0, -8)\); \((1,0),\;(−1,0)\)

    51. \((0,0),\;(\sqrt{3},0)),\;(−\sqrt{3},0)\)

    53. \((0,0),\,(1,0),(−1,0),(2,0),(−2,0)\)

    55. \((0, 0)\); \((0, 0), (2, 0)\).  As \(x→\pm ∞,\) \(f(x)→∞\)

    57. \((0,0)\);  \((0, 0), (5, 0), (7, 0)\). As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

    59. \((0, 0)\); \((−4, 0), (0, 0), (4, 0)\). As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

    61. \((0, -81)\); \((3, 0), (−3, 0)\). As \(x→\pm ∞,\) \(f(x)→∞\)

    63.\((0, 0)\); \((−3, 0), (0, 0), (5, 0).\) As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

    \( \star \)


    3.3e: Exercises - Polynomial End Behaviour is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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