{"id":2394,"date":"2019-05-14T13:43:36","date_gmt":"2019-05-14T13:43:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2394"},"modified":"2019-05-14T13:43:36","modified_gmt":"2019-05-14T13:43:36","slug":"4-1-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/4-1-section-exercises\/","title":{"raw":"4.1 Section Exercises","rendered":"4.1 Section Exercises"},"content":{"raw":"
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4.1 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. Explain the difference between the coefficient of a power function and its degree.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137423709\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137423709\"]\r\n

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.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n2. If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?\r\n\r\n<\/div>\r\n<\/div>\r\n
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3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135205853\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135205853\"]\r\n

As[latex]\\text{ }x\\text{ }[\/latex] decreases without bound, so does[latex]\\text{ }f\\left(x\\right).\\text{ }[\/latex] As[latex]\\text{ }x\\text{ }[\/latex] increases without bound, so does[latex]\\text{ }f\\left(x\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n5. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As[latex]\\text{ }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to -\\infty \\text{ }[\/latex] and as[latex]\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty .\\text{ }[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137679007\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137679007\"]\r\n

The polynomial function is of even degree and leading coefficient is negative.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Algebraic<\/h4>\r\n

For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\r\n\r\n

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6. [latex]f\\left(x\\right)={x}^{5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7. [latex]f\\left(x\\right)={\\left({x}^{2}\\right)}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134043753\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134043753\"]\r\n

Power function<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. [latex]f\\left(x\\right)=x-{x}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137723151\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137723151\"]\r\n

Neither<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. [latex]f\\left(x\\right)=2x\\left(x+2\\right){\\left(x-1\\right)}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [latex]f\\left(x\\right)={3}^{x+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137619578\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137619578\"]\r\n

Neither<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the degree and leading coefficient for the given polynomial.<\/p>\r\n\r\n

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12. [latex]-3x{}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [latex]7-2{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137424474\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137424474\"]\r\n

Degree = 2, Coefficient = \u20132<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [latex]-2{x}^{2}- 3{x}^{5}+ x-6 [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [latex]x\\left(4-{x}^{2}\\right)\\left(2x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137842483\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137842483\"]\r\n

Degree =4, Coefficient = \u20132<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [latex]{x}^{2}{\\left(2x-3\\right)}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, determine the end behavior of the functions.<\/p>\r\n\r\n

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17. [latex]f\\left(x\\right)={x}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137445856\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137445856\"]\r\n

[latex]\\text{As}\\text{ }x\\to \\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to \\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18. [latex]f\\left(x\\right)={x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19. [latex]f\\left(x\\right)=-{x}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137643430\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137643430\"]\r\n

[latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. [latex]f\\left(x\\right)=-{x}^{9}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]f\\left(x\\right)=-2{x}^{4}- 3{x}^{2}+ x-1 [\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137656888\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137656888\"]\r\n

[latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. [latex]f\\left(x\\right)=3{x}^{2}+ x-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [latex]f\\left(x\\right)={x}^{2}\\left(2{x}^{3}-x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137794123\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137794123\"]\r\n

[latex]\\text{As}\\text{ }x\\to \\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24. [latex]f\\left(x\\right)={\\left(2-x\\right)}^{7}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the intercepts of the functions.<\/p>\r\n\r\n

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25. [latex]f\\left(t\\right)=2\\left(t-1\\right)\\left(t+2\\right)\\left(t-3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135697918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135697918\"]\r\n

y<\/em>-intercept is[latex]\\text{ }\\left(0,12\\right),\\text{ }[\/latex]t<\/em>-intercepts are[latex]\\text{ }\\left(1,0\\right);\\left(\u20132,0\\right);\\text{and }\\left(3,0\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. [latex]g\\left(n\\right)=-2\\left(3n-1\\right)\\left(2n+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. [latex]f\\left(x\\right)={x}^{4}-16[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137728286\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137728286\"]\r\n

y<\/em>-intercept is[latex]\\text{ }\\left(0,-16\\right).\\text{ }[\/latex]x<\/em>-intercepts are[latex]\\text{ }\\left(2,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(-2,0\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28. [latex]f\\left(x\\right)={x}^{3}+27[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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29. [latex]f\\left(x\\right)=x\\left({x}^{2}-2x-8\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137896961\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137896961\"]\r\n

y<\/em>-intercept is[latex]\\text{ }\\left(0,0\\right).\\text{ }[\/latex]x<\/em>-intercepts are[latex]\\text{ }\\left(0,0\\right),\\left(4,0\\right),\\text{ }[\/latex]and[latex]\\text{ }\\left(-2, 0\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. [latex]f\\left(x\\right)=\\left(x+3\\right)\\left(4{x}^{2}-1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Graphical<\/h4>\r\n

For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\r\n\r\n

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31.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135388469\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135388469\"]\r\n

3<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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33.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135181683\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135181683\"]\r\n

5<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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35.\u00a0\"Graph<\/span><\/div>\r\n
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[reveal-answer q=\"580167\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"580167\"]<\/p>\r\n3[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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36.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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37.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135191388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135191388\"]\r\n

5<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/p>\r\n\r\n

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39.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137444866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137444866\"]\r\n

Yes. Number of turning points is 2. Least possible degree is 3.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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41.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137570516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137570516\"]\r\n

Yes. Number of turning points is 1. Least possible degree is 2.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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42.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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43.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134042451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134042451\"]\r\n

Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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44.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165133310452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133310452\"]\r\n

No.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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45.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137569703\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137569703\"]\r\n

Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Numeric<\/h4>\r\n

For the following exercises, make a table to confirm the end behavior of the function.<\/p>\r\n\r\n

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46. [latex]f\\left(x\\right)=-{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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47. [latex]f\\left(x\\right)={x}^{4}-5{x}^{2}[\/latex]<\/p>\r\n[reveal-answer q=\"318948\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"318948\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a0<\/caption>\r\n
[latex]x[\/latex]<\/th>\r\n[latex]f\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
10<\/td>\r\n9,500<\/td>\r\n<\/tr>\r\n
100<\/td>\r\n99,950,000<\/td>\r\n<\/tr>\r\n
\u201310<\/td>\r\n9,500<\/td>\r\n<\/tr>\r\n
\u2013100<\/td>\r\n99,950,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n[latex]\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty [\/latex]\r\n
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48. [latex]f\\left(x\\right)={x}^{2}{\\left(1-x\\right)}^{2}[\/latex]<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. [latex]f\\left(x\\right)=\\left(x-1\\right)\\left(x-2\\right)\\left(3-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134122928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134122928\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a0<\/caption>\r\n
[latex]x[\/latex]<\/th>\r\n[latex]f\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
10<\/td>\r\n\u2013504<\/td>\r\n<\/tr>\r\n
100<\/td>\r\n\u2013941,094<\/td>\r\n<\/tr>\r\n
\u201310<\/td>\r\n1,716<\/td>\r\n<\/tr>\r\n
\u2013100<\/td>\r\n1,061,106<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

[latex]\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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50. [latex]f\\left(x\\right)=\\frac{{x}^{5}}{10}-{x}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Technology<\/h4>\r\n

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/p>\r\n\r\n

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51. [latex]f\\left(x\\right)={x}^{3}\\left(x-2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137629480\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137629480\"]\"Graph<\/span>\r\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0, 0\\right).\\text{ }[\/latex] The[latex]\\text{ }x\\text{-}[\/latex]intercepts are[latex]\\text{ }\\left(0, 0\\right),\\text{ }\\left(2, 0\\right).\\text{ }[\/latex] [latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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52. [latex]f\\left(x\\right)=x\\left(x-3\\right)\\left(x+3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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53. [latex]f\\left(x\\right)=x\\left(14-2x\\right)\\left(10-2x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135436598\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135436598\"]\"Graph<\/span>\r\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0,0\\right)[\/latex] . The[latex]\\text{ }x\\text{-}[\/latex]intercepts are[latex]\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right),\\text{ }\\left(7, 0\\right).\\text{ }[\/latex] [latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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54. [latex]f\\left(x\\right)=x\\left(14-2x\\right){\\left(10-2x\\right)}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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55. [latex]f\\left(x\\right)={x}^{3}-16x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137676496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137676496\"]\"\"<\/span>\r\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0, 0\\right).\\text{ }[\/latex] The[latex]\\text{ }x\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(-4, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(4, 0\\right).\\text{ }[\/latex] [latex]As\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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56. [latex]f\\left(x\\right)={x}^{3}-27[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. [latex]f\\left(x\\right)={x}^{4}-81[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165131891798\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165131891798\"]\"Graph<\/span>\r\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0, -81\\right).\\text{ }[\/latex] The[latex]\\text{ }x\\text{-}[\/latex]intercept are[latex]\\text{ }\\left(3, 0\\right),\\text{ }\\left(-3, 0\\right).\\text{ }[\/latex] [latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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58. [latex]f\\left(x\\right)=-{x}^{3}+{x}^{2}+2x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59. [latex]f\\left(x\\right)={x}^{3}-2{x}^{2}-15x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134381574\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134381574\"]\"Graph<\/span>\r\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0, 0\\right).\\text{ }[\/latex] The[latex]\\text{ }x\\text{-}[\/latex]intercepts are[latex]\\text{ }\\left(-3, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right).\\text{ }[\/latex] [latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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60. [latex]f\\left(x\\right)={x}^{3}-0.01x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Extensions<\/h4>\r\n

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.<\/p>\r\n\r\n

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61. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,-4\\right).\\text{ }[\/latex]The[latex]\\text{ }x-[\/latex]intercepts are[latex]\\text{ }\\left(-2,0\\right),\\text{ }\\left(2,0\\right).\\text{ }[\/latex]Degree is 2.<\/p>\r\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135160156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135160156\"]\r\n

[latex]f\\left(x\\right)={x}^{2}-4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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62. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,9\\right).\\text{ }[\/latex]The[latex]\\text{ }x\\text{-}[\/latex]intercepts are[latex]\\text{ }\\left(-3,0\\right),\\text{ }\\left(3,0\\right).\\text{ }[\/latex]Degree is 2.<\/p>\r\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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63. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,0\\right).\\text{ }[\/latex]The[latex]\\text{ }x-[\/latex]intercepts are[latex]\\text{ }\\left(0,0\\right),\\text{ }\\left(2,0\\right).\\text{ }[\/latex]Degree is 3.<\/p>\r\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134389974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134389974\"]\r\n

[latex]f\\left(x\\right)={x}^{3}-4{x}^{2}+4x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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64. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,1\\right).\\text{ }[\/latex]The[latex]\\text{ }x-[\/latex]intercept is[latex]\\text{ }\\left(1,0\\right).\\text{ }[\/latex]Degree is 3.<\/p>\r\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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65. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,1\\right).\\text{ }[\/latex]There is no[latex]\\text{ }x-[\/latex]intercept. Degree is 4.<\/p>\r\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137922537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137922537\"]\r\n

[latex]f\\left(x\\right)={x}^{4}+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Real-World Applications<\/h4>\r\n

For the following exercises, use the written statements to construct a polynomial function that represents the required information.<\/p>\r\n\r\n

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66. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of[latex]\\text{ }d,\\text{ }[\/latex]the number of days elapsed.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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67. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of[latex]\\text{ }m,\\text{ }[\/latex]the number of minutes elapsed.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134032299\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134032299\"]\r\n

[latex]V\\left(m\\right)=8{m}^{3}+36{m}^{2}+54m+27[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by[latex]\\text{ }x\\text{ }[\/latex]inches and the width increased by twice that amount, express the area of the rectangle as a function of[latex]\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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69. An open box is to be constructed by cutting out square corners of [latex]\\text{ }x-[\/latex]inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of[latex]\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135571779\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135571779\"]\r\n

[latex]V\\left(x\\right)=4{x}^{3}-32{x}^{2}+64x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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70. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width ([latex]x[\/latex]).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

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4.1 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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1. Explain the difference between the coefficient of a power function and its degree.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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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?<\/p>\n<\/div>\n<\/div>\n

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3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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As[latex]\\text{ }x\\text{ }[\/latex] decreases without bound, so does[latex]\\text{ }f\\left(x\\right).\\text{ }[\/latex] As[latex]\\text{ }x\\text{ }[\/latex] increases without bound, so does[latex]\\text{ }f\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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4. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?<\/p>\n<\/div>\n<\/div>\n

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5. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As[latex]\\text{ }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to -\\infty \\text{ }[\/latex] and as[latex]\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty .\\text{ }[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The polynomial function is of even degree and leading coefficient is negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n

For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\n

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6. [latex]f\\left(x\\right)={x}^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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7. [latex]f\\left(x\\right)={\\left({x}^{2}\\right)}^{3}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Power function<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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8. [latex]f\\left(x\\right)=x-{x}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9. [latex]f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}-1}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Neither<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10. [latex]f\\left(x\\right)=2x\\left(x+2\\right){\\left(x-1\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11. [latex]f\\left(x\\right)={3}^{x+1}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Neither<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the degree and leading coefficient for the given polynomial.<\/p>\n

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12. [latex]-3x{}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13. [latex]7-2{x}^{2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Degree = 2, Coefficient = \u20132<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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14. [latex]-2{x}^{2}- 3{x}^{5}+ x-6[\/latex]<\/p>\n<\/div>\n<\/div>\n

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15. [latex]x\\left(4-{x}^{2}\\right)\\left(2x+1\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

Degree =4, Coefficient = \u20132<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16. [latex]{x}^{2}{\\left(2x-3\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, determine the end behavior of the functions.<\/p>\n

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17. [latex]f\\left(x\\right)={x}^{4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\text{As}\\text{ }x\\to \\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18. [latex]f\\left(x\\right)={x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19. [latex]f\\left(x\\right)=-{x}^{4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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20. [latex]f\\left(x\\right)=-{x}^{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21. [latex]f\\left(x\\right)=-2{x}^{4}- 3{x}^{2}+ x-1[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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22. [latex]f\\left(x\\right)=3{x}^{2}+ x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23. [latex]f\\left(x\\right)={x}^{2}\\left(2{x}^{3}-x+1\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\text{As}\\text{ }x\\to \\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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24. [latex]f\\left(x\\right)={\\left(2-x\\right)}^{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, find the intercepts of the functions.<\/p>\n

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25. [latex]f\\left(t\\right)=2\\left(t-1\\right)\\left(t+2\\right)\\left(t-3\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

y<\/em>-intercept is[latex]\\text{ }\\left(0,12\\right),\\text{ }[\/latex]t<\/em>-intercepts are[latex]\\text{ }\\left(1,0\\right);\\left(\u20132,0\\right);\\text{and }\\left(3,0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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26. [latex]g\\left(n\\right)=-2\\left(3n-1\\right)\\left(2n+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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27. [latex]f\\left(x\\right)={x}^{4}-16[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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y<\/em>-intercept is[latex]\\text{ }\\left(0,-16\\right).\\text{ }[\/latex]x<\/em>-intercepts are[latex]\\text{ }\\left(2,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(-2,0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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28. [latex]f\\left(x\\right)={x}^{3}+27[\/latex]<\/p>\n<\/div>\n<\/div>\n

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29. [latex]f\\left(x\\right)=x\\left({x}^{2}-2x-8\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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y<\/em>-intercept is[latex]\\text{ }\\left(0,0\\right).\\text{ }[\/latex]x<\/em>-intercepts are[latex]\\text{ }\\left(0,0\\right),\\left(4,0\\right),\\text{ }[\/latex]and[latex]\\text{ }\\left(-2, 0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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30. [latex]f\\left(x\\right)=\\left(x+3\\right)\\left(4{x}^{2}-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Graphical<\/h4>\n

For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\n

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31.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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33.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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35.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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3<\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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36.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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37.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38.\u00a0\"Graph<\/span><\/div>\n<\/div>\n

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/p>\n

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39.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
\n

Yes. Number of turning points is 2. Least possible degree is 3.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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40.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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41.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
\n

Yes. Number of turning points is 1. Least possible degree is 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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42.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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43.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
\n

Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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44.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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No.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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45.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
\n

Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Numeric<\/h4>\n

For the following exercises, make a table to confirm the end behavior of the function.<\/p>\n

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46. [latex]f\\left(x\\right)=-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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47. [latex]f\\left(x\\right)={x}^{4}-5{x}^{2}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\n\n\n\n\n\n\n\n\n
\u00a0<\/caption>\n
[latex]x[\/latex]<\/th>\n[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
10<\/td>\n9,500<\/td>\n<\/tr>\n
100<\/td>\n99,950,000<\/td>\n<\/tr>\n
\u201310<\/td>\n9,500<\/td>\n<\/tr>\n
\u2013100<\/td>\n99,950,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n

[latex]\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n

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48. [latex]f\\left(x\\right)={x}^{2}{\\left(1-x\\right)}^{2}[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

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49. [latex]f\\left(x\\right)=\\left(x-1\\right)\\left(x-2\\right)\\left(3-x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\u00a0<\/caption>\n
[latex]x[\/latex]<\/th>\n[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
10<\/td>\n\u2013504<\/td>\n<\/tr>\n
100<\/td>\n\u2013941,094<\/td>\n<\/tr>\n
\u201310<\/td>\n1,716<\/td>\n<\/tr>\n
\u2013100<\/td>\n1,061,106<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

[latex]\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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50. [latex]f\\left(x\\right)=\\frac{{x}^{5}}{10}-{x}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Technology<\/h4>\n

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/p>\n

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51. [latex]f\\left(x\\right)={x}^{3}\\left(x-2\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"Graph<\/span><\/p>\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0, 0\\right).\\text{ }[\/latex] The[latex]\\text{ }x\\text{-}[\/latex]intercepts are[latex]\\text{ }\\left(0, 0\\right),\\text{ }\\left(2, 0\\right).\\text{ }[\/latex] [latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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52. [latex]f\\left(x\\right)=x\\left(x-3\\right)\\left(x+3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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53. [latex]f\\left(x\\right)=x\\left(14-2x\\right)\\left(10-2x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"Graph<\/span><\/p>\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0,0\\right)[\/latex] . The[latex]\\text{ }x\\text{-}[\/latex]intercepts are[latex]\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right),\\text{ }\\left(7, 0\\right).\\text{ }[\/latex] [latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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54. [latex]f\\left(x\\right)=x\\left(14-2x\\right){\\left(10-2x\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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55. [latex]f\\left(x\\right)={x}^{3}-16x[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"\"<\/span><\/p>\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0, 0\\right).\\text{ }[\/latex] The[latex]\\text{ }x\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(-4, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(4, 0\\right).\\text{ }[\/latex] [latex]As\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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56. [latex]f\\left(x\\right)={x}^{3}-27[\/latex]<\/p>\n<\/div>\n<\/div>\n

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57. [latex]f\\left(x\\right)={x}^{4}-81[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"Graph<\/span><\/p>\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0, -81\\right).\\text{ }[\/latex] The[latex]\\text{ }x\\text{-}[\/latex]intercept are[latex]\\text{ }\\left(3, 0\\right),\\text{ }\\left(-3, 0\\right).\\text{ }[\/latex] [latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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58. [latex]f\\left(x\\right)=-{x}^{3}+{x}^{2}+2x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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59. [latex]f\\left(x\\right)={x}^{3}-2{x}^{2}-15x[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"Graph<\/span><\/p>\n

The[latex]\\text{ }y\\text{-}[\/latex]intercept is[latex]\\text{ }\\left(0, 0\\right).\\text{ }[\/latex] The[latex]\\text{ }x\\text{-}[\/latex]intercepts are[latex]\\text{ }\\left(-3, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right).\\text{ }[\/latex] [latex]\\text{As}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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60. [latex]f\\left(x\\right)={x}^{3}-0.01x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Extensions<\/h4>\n

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.<\/p>\n

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61. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,-4\\right).\\text{ }[\/latex]The[latex]\\text{ }x-[\/latex]intercepts are[latex]\\text{ }\\left(-2,0\\right),\\text{ }\\left(2,0\\right).\\text{ }[\/latex]Degree is 2.<\/p>\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)={x}^{2}-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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62. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,9\\right).\\text{ }[\/latex]The[latex]\\text{ }x\\text{-}[\/latex]intercepts are[latex]\\text{ }\\left(-3,0\\right),\\text{ }\\left(3,0\\right).\\text{ }[\/latex]Degree is 2.<\/p>\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n

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63. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,0\\right).\\text{ }[\/latex]The[latex]\\text{ }x-[\/latex]intercepts are[latex]\\text{ }\\left(0,0\\right),\\text{ }\\left(2,0\\right).\\text{ }[\/latex]Degree is 3.<\/p>\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to -\\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)={x}^{3}-4{x}^{2}+4x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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64. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,1\\right).\\text{ }[\/latex]The[latex]\\text{ }x-[\/latex]intercept is[latex]\\text{ }\\left(1,0\\right).\\text{ }[\/latex]Degree is 3.<\/p>\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to -\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n

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65. The[latex]\\text{ }y-[\/latex]intercept is[latex]\\text{ }\\left(0,1\\right).\\text{ }[\/latex]There is no[latex]\\text{ }x-[\/latex]intercept. Degree is 4.<\/p>\n

End behavior:[latex]\\text{ }\\text{as}\\text{ }x\\to -\\infty ,\\text{ }\\text{ }f\\left(x\\right)\\to \\infty ,\\text{ }\\text{as}\\text{ }x\\to \\infty ,\\text{ }f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

[latex]f\\left(x\\right)={x}^{4}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Real-World Applications<\/h4>\n

For the following exercises, use the written statements to construct a polynomial function that represents the required information.<\/p>\n

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66. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of[latex]\\text{ }d,\\text{ }[\/latex]the number of days elapsed.<\/p>\n<\/div>\n<\/div>\n

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67. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of[latex]\\text{ }m,\\text{ }[\/latex]the number of minutes elapsed.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]V\\left(m\\right)=8{m}^{3}+36{m}^{2}+54m+27[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by[latex]\\text{ }x\\text{ }[\/latex]inches and the width increased by twice that amount, express the area of the rectangle as a function of[latex]\\text{ }x.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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69. An open box is to be constructed by cutting out square corners of [latex]\\text{ }x-[\/latex]inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of[latex]\\text{ }x.[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]V\\left(x\\right)=4{x}^{3}-32{x}^{2}+64x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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70. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width ([latex]x[\/latex]).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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