Graphs of Polynomials - Revision history

Khanh at 21:00, 5 November 2021

2021-11-05T21:00:19Z

Khanh at 20:59, 5 November 2021

2021-11-05T20:59:24Z

Khanh at 20:57, 5 November 2021

2021-11-05T20:57:28Z

Lila at 19:39, 21 October 2021

2021-10-21T19:39:14Z

Lila: /* Graphs */

2021-10-21T19:38:16Z

Graphs

Lila at 19:35, 21 October 2021

2021-10-21T19:35:50Z

Lila at 19:33, 21 October 2021

2021-10-21T19:33:08Z

Lila: /* Resources */

2021-10-21T19:31:46Z

Resources

Lila: /* Resources */

2021-10-21T19:30:59Z

Resources

Lila at 19:30, 21 October 2021

2021-10-21T19:30:02Z

@@ Line 8: / Line 8: @@
 File:Quintic polynomial.svg|Polynomial of degree 5:<br/>{{math|1=''f''(''x'') = 1/20 (''x'' + 4)(''x'' + 2)(''x'' + 1)(''x'' − 1)<br/>(''x'' − 3) + 2}}
 File:Sextic Graph.svg|Polynomial of degree 6:<br/>{{math|1=''f''(''x'') = 1/100 (''x''<sup>6</sup> − 2''x'' <sup>5</sup> − 26''x''<sup>4</sup> + 28''x''<sup>3</sup>}}<br/>{{math|+ 145''x''<sup>2</sup> − 26''x'' − 80)}}
-File:Septic graph.svg|Polynomial of degree 7:<br/>{{math|1=''f''(''x'') = (''x'' − 3)(''x'' − 2)(''x'' − 1)(''x'')(''x'' + 1)(''x'' + 2)(''x'' + 3)}}
+File:Septic graph.svg|Polynomial of degree 7:<br/>{{math|1=''f''(''x'') = (''x'' − 3)(''x'' − 2)(''x'' − 1)(''x'')(''x'' + 1)<br/>(''x'' + 2)(''x'' + 3)}}
 </gallery>
 </div>

@@ Line 8: / Line 8: @@
 File:Quintic polynomial.svg|Polynomial of degree 5:<br/>{{math|1=''f''(''x'') = 1/20 (''x'' + 4)(''x'' + 2)(''x'' + 1)(''x'' − 1)<br/>(''x'' − 3) + 2}}
 File:Sextic Graph.svg|Polynomial of degree 6:<br/>{{math|1=''f''(''x'') = 1/100 (''x''<sup>6</sup> − 2''x'' <sup>5</sup> − 26''x''<sup>4</sup> + 28''x''<sup>3</sup>}}<br/>{{math|+ 145''x''<sup>2</sup> − 26''x'' − 80)}}
-File:Septic graph.svg|Polynomial of degree 7:<br/>{{math|1=''f''(''x'') = (''x'' − 3)(''x'' − 2)(''x'' − 1)(''x'')(''x'' + 1)(''x'' + 2)}}<br/>{{math|(''x'' + 3)}}
+File:Septic graph.svg|Polynomial of degree 7:<br/>{{math|1=''f''(''x'') = (''x'' − 3)(''x'' − 2)(''x'' − 1)(''x'')(''x'' + 1)(''x'' + 2)(''x'' + 3)}}
 </gallery>
 </div>

@@ Line 3: / Line 3: @@
 File:Algebra1 fnz fig037 pc.svg|Polynomial of degree 0:<br/>{{math|1=''f''(''x'') = 2}}
 File:Fonction de Sophie Germain.png|Polynomial of degree 1:<br/>{{math|1=''f''(''x'') = 2''x'' + 1}}
-File:Polynomialdeg2.svg|Polynomial of degree 2:<br/>{{math|1=''f''(''x'') = ''x''<sup>2</sup> − ''x'' − 2}}<br/>{{math|= (''x'' + 1)(''x'' − 2)}}
+File:Polynomialdeg2.svg|Polynomial of degree 2:<br/>{{math|1=''f''(''x'') = ''x''<sup>2</sup> − ''x'' − 2}}<br/>= (''x'' + 1)(''x'' − 2)
-File:Polynomialdeg3.svg|Polynomial of degree 3:<br/>{{math|1=''f''(''x'') = ''x''<sup>3</sup>/4 + 3''x''<sup>2</sup>/4 − 3''x''/2 − 2}}<br/>{{math|= 1/4 (''x'' + 4)(''x'' + 1)(''x'' − 2)}}
+File:Polynomialdeg3.svg|Polynomial of degree 3:<br/>{{math|1=''f''(''x'') = ''x''<sup>3</sup>/4 + 3''x''<sup>2</sup>/4 − 3''x''/2 − 2}}<br/>= 1/4 (''x'' + 4)(''x'' + 1)(''x'' − 2)
 File:Polynomialdeg4.svg|Polynomial of degree 4:<br/>{{math|1=''f''(''x'') = 1/14 (''x'' + 4)(''x'' + 1)(''x'' − 1)(''x'' − 3) <br/>+ 0.5}}
 File:Quintic polynomial.svg|Polynomial of degree 5:<br/>{{math|1=''f''(''x'') = 1/20 (''x'' + 4)(''x'' + 2)(''x'' + 1)(''x'' − 1)<br/>(''x'' − 3) + 2}}

@@ Line 13: / Line 13: @@
-A ''polynomial function'' is a function that can be defined by [[#evaluation|evaluating]] a polynomial. More precisely, a function {{math|''f''}} of one argument from a given domain is a polynomial function if there exists a polynomial
+A ''polynomial function'' is a function that can be defined by evaluating a polynomial. More precisely, a function {{math|''f''}} of one argument from a given domain is a polynomial function if there exists a polynomial
 :<math>a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 </math>
 that evaluates to <math>f(x)</math> for all {{mvar|x}} in the domain of {{mvar|f}} (here, {{math|''n''}} is a non-negative integer and {{math|''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a<sub>n</sub>''}} are constant coefficients).

@@ Line 32: / Line 32: @@
 <li>
 The graph of the zero polynomial
-{{block indent|{{math|1=''f''(''x'') = 0}}}} is the {{math|''x''}}-axis.
+{{math|1=''f''(''x'') = 0}} is the {{math|''x''}}-axis.
 </li>
 <li>
 The graph of a degree 0 polynomial
-{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub>}}, where {{math|''a''<sub>0</sub> ≠ 0}},}} is a horizontal line with {{nowrap|{{math|''y''}}-intercept {{math|''a''<sub>0</sub>}}}}
+{{math|1=''f''(''x'') = ''a''<sub>0</sub>}}, where {{math|''a''<sub>0</sub> ≠ 0}}, is a horizontal line with {{math|''y''}}-intercept {{math|''a''<sub>0</sub>}}
 </li>
 <li>
 The graph of a degree 1 polynomial (or linear function)
-{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x''}}, where {{math|''a''<sub>1</sub> ≠ 0}},}} is an oblique line with {{nowrap|{{math|''y''}}-intercept {{math|''a''<sub>0</sub>}}}} and [[slope]] {{math|''a''<sub>1</sub>}}.
+{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x''}}, where {{math|''a''<sub>1</sub> ≠ 0}}, is an oblique line with {{math|''y''}}-intercept {{math|''a''<sub>0</sub>}} and slope {{math|''a''<sub>1</sub>}}.
 </li>
 <li>
 The graph of a degree 2 polynomial
-{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup>}}, where {{math|''a''<sub>2</sub> ≠ 0}}}} is a [[parabola]].
+{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup>}}, where {{math|''a''<sub>2</sub> ≠ 0}} is a parabola.
 </li>
 <li>
 The graph of a degree 3 polynomial
-{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup> + ''a''<sub>3</sub>''x''<sup>3</sup>}}, where {{math|''a''<sub>3</sub> ≠ 0}}}} is a [[cubic equation|cubic curve]].
+{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup> + ''a''<sub>3</sub>''x''<sup>3</sup>}}, where {{math|''a''<sub>3</sub> ≠ 0}} is a cubic curve.
 </li>
 <li>
 The graph of any polynomial with degree 2 or greater
-{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup> + ⋯ + ''a''<sub>''n''</sub>''x''<sup>''n''</sup>}}, where {{math|''a''<sub>''n''</sub> ≠ 0 and ''n'' ≥ 2}}}} is a continuous non-linear curve.
+{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup> + ⋯ + ''a''<sub>''n''</sub>''x''<sup>''n''</sup>}}, where {{math|''a''<sub>''n''</sub> ≠ 0 and ''n'' ≥ 2}} is a continuous non-linear curve.
 </li>
 </ul>

@@ Line 1: / Line 1: @@
 <div class="floatright">
 <gallery perrow="2" widths="200px" heights="200px">
-File:Algebra1 fnz fig037 pc.svg|Polynomial of degree 0:<br/><small>{{math|1=''f''(''x'') = 2}}</small>
+File:Algebra1 fnz fig037 pc.svg|Polynomial of degree 0:<br/>{{math|1=''f''(''x'') = 2}}
-File:Fonction de Sophie Germain.png|Polynomial of degree 1:<br/><small>{{math|''f''(''x'') = 2''x'' + 1}}</small>
+File:Fonction de Sophie Germain.png|Polynomial of degree 1:<br/>{{math|1=''f''(''x'') = 2''x'' + 1}}
-File:Polynomialdeg2.svg|Polynomial of degree 2:<br/><small>{{math|''f''(''x'') = ''x''<sup>2</sup> − ''x'' − 2}}<br/>{{math|= (''x'' + 1)(''x'' − 2)}}</small>
+File:Polynomialdeg2.svg|Polynomial of degree 2:<br/>{{math|1=''f''(''x'') = ''x''<sup>2</sup> − ''x'' − 2}}<br/>{{math|= (''x'' + 1)(''x'' − 2)}}
-File:Polynomialdeg3.svg|Polynomial of degree 3:<br/><small>{{math|''f''(''x'') = ''x''<sup>3</sup>/4 + 3''x''<sup>2</sup>/4 − 3''x''/2 − 2}}<br/>{{math|= 1/4 (''x'' + 4)(''x'' + 1)(''x'' − 2)}}</small>
+File:Polynomialdeg3.svg|Polynomial of degree 3:<br/>{{math|1=''f''(''x'') = ''x''<sup>3</sup>/4 + 3''x''<sup>2</sup>/4 − 3''x''/2 − 2}}<br/>{{math|= 1/4 (''x'' + 4)(''x'' + 1)(''x'' − 2)}}
-File:Polynomialdeg4.svg|Polynomial of degree 4:<br/><small>{{math|''f''(''x'') = 1/14 (''x'' + 4)(''x'' + 1)(''x'' − 1)(''x'' − 3) <br/>+ 0.5}}</small>
+File:Polynomialdeg4.svg|Polynomial of degree 4:<br/>{{math|1=''f''(''x'') = 1/14 (''x'' + 4)(''x'' + 1)(''x'' − 1)(''x'' − 3) <br/>+ 0.5}}
-File:Quintic polynomial.svg|Polynomial of degree 5:<br/><small>{{math|''f''(''x'') = 1/20 (''x'' + 4)(''x'' + 2)(''x'' + 1)(''x'' − 1)<br/>(''x'' − 3) + 2}}</small>
+File:Quintic polynomial.svg|Polynomial of degree 5:<br/>{{math|1=''f''(''x'') = 1/20 (''x'' + 4)(''x'' + 2)(''x'' + 1)(''x'' − 1)<br/>(''x'' − 3) + 2}}
-File:Sextic Graph.svg|Polynomial of degree 6:<br/><small>{{math|''f''(''x'') = 1/100 (''x''<sup>6</sup> − 2''x'' <sup>5</sup> − 26''x''<sup>4</sup> + 28''x''<sup>3</sup>}}<br/>{{math|+ 145''x''<sup>2</sup> − 26''x'' − 80)}}</small>
+File:Sextic Graph.svg|Polynomial of degree 6:<br/>{{math|1=''f''(''x'') = 1/100 (''x''<sup>6</sup> − 2''x'' <sup>5</sup> − 26''x''<sup>4</sup> + 28''x''<sup>3</sup>}}<br/>{{math|+ 145''x''<sup>2</sup> − 26''x'' − 80)}}
-File:Septic graph.svg|Polynomial of degree 7:<br/><small>{{math|''f''(''x'') = (''x'' − 3)(''x'' − 2)(''x'' − 1)(''x'')(''x'' + 1)(''x'' + 2)}}<br/>{{math|(''x'' + 3)}}</small>
+File:Septic graph.svg|Polynomial of degree 7:<br/>{{math|1=''f''(''x'') = (''x'' − 3)(''x'' − 2)(''x'' − 1)(''x'')(''x'' + 1)(''x'' + 2)}}<br/>{{math|(''x'' + 3)}}
 </gallery>
 </div>
-A ''polynomial function'' is a function that can be defined by [[#evaluation|evaluating]] a polynomial. More precisely, a function {{math|''f''}} of one [[argument of a function|argument]] from a given domain is a polynomial function if there exists a polynomial
 :<math>a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 </math>
-that evaluates to <math>f(x)</math> for all {{mvar|x}} in the [[domain of a function|domain]] of {{mvar|f}} (here, {{math|''n''}} is a non-negative integer and {{math|''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a<sub>n</sub>''}} are constant coefficients).
+that evaluates to <math>f(x)</math> for all {{mvar|x}} in the domain of {{mvar|f}} (here, {{math|''n''}} is a non-negative integer and {{math|''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a<sub>n</sub>''}} are constant coefficients).
-Generally, unless otherwise specified, polynomial functions have [[complex number|complex]] coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also [[restriction of a function|restricted]] to the reals, the resulting function is a [[real function]] that maps reals to reals.
+Generally, unless otherwise specified, polynomial functions have complex coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also restricted to the reals, the resulting function is a real function that maps reals to reals.
 For example, the function {{math|''f''}}, defined by
@@ Line 24: / Line 25: @@
 According to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression <math>\left(\sqrt{1-x^2}\right)^2,</math> which takes the same values as the polynomial <math>1-x^2</math> on the interval <math>[-1,1]</math>, and thus both expressions define the same polynomial function on this interval.
-Every polynomial function is [[continuous function|continuous]], [[smooth function|smooth]], and [[entire function|entire]].
+Every polynomial function is continuous, smooth, and entire.
 ===Graphs===
-A polynomial function in one real variable can be represented by a [[graph of a function|graph]].
+A polynomial function in one real variable can be represented by a graph.
 <ul>
 <li>
@@ Line 55: / Line 56: @@
 </ul>
-A non-constant polynomial function [[infinity#Calculus|tends to infinity]] when the variable increases indefinitely (in [[absolute value]]). If the degree is higher than one, the graph does not have any [[asymptote]]. It has two [[parabolic branch]]es with vertical direction (one branch for positive ''x'' and one for negative ''x'').
+A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value). If the degree is higher than one, the graph does not have any asymptote. It has two parabolic branches with vertical direction (one branch for positive ''x'' and one for negative ''x'').
 Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

@@ Line 60: / Line 60: @@
 ==Resources==
 * [https://courses.lumenlearning.com/collegealgebra2017/chapter/graphs-of-polynomial-functions/ Graphs of Polynomial Functions], Lumen Learning College Algebra
 * [https://en.wikipedia.org/wiki/Polynomial Polynomial], Wikipedia