How To Calculate The Polynomials Of A Polynomial Function
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- Mervyn Parrish
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1 1. Sketch the grapf of the following polynomial function: f. Sketch the grapf of the following polynomial function: f. Sketch the grapf of the following polynomial function: f 4. Sketch the grapf of the following polynomial function: f 5. Sketch the grapf of the following polynomial function: f. Sketch the grapf of the following polynomial function: f 7. Sketch the grapf of the following polynomial function: f 8. Sketch the grapf of the following polynomial function: f 9. Sketch the grapf of the following polynomial function: f 10. Sketch the grapf of the following polynomial function: f (x + )(x + 1)(x ) (x + )(x 1)(x ) (x + 1)(x 0)(x ) (x + )(x + )(x ) (x + 1)(x )(x ) (x + )(x + 1)(x ) (x + )(x 0)(x 1) (x + 1)(x 1)(x ) (x + 1)(x )(x ) (x + )(x + )(x + 1) c 009 La Citadelle 1 of 1
2 Solutions: (x + )(x + 1)(x ) 1. f By expanding, f 9 9x + x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = 9 9( x) + ( x) + ( x) f( x) f(x) f( x) f(x) = 9 + 9x + x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = x = 1 x = y-intercept y int = f(0) = 9 =.50 Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f 9 + x + x Critical numbers are the solutions of the equation f 0 or 9 + x + x = 0 x = ± ( ) 4()( 9) x 4 = (11) =.097 y 4 = f(x 4 ) = 1. x 5 = + (11) = 1.41 y 5 = f(x 5 ) = 4.5 Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is decreasing over (,.097) and over (1.41, ) and is increasing over (.097, 1.41). Maximum and Minimum Points The function f(x) has a minimum point at P 4 (.097, 1.) and a maximum point at P 5 (1.41, 4.5). Concavity Intervals The second derivative of the function f(x) is given by: f + x c 009 La Citadelle of 1
3 The second derivative f (x) is zero when: f 0 + x = 0 x = 1 Sign Chart for the Second Derivative f (x) y = f(x ) = x 0. f(x) f (x) + 0 Inflection Points There is an inflection point at P = ( 0., 1.481) Graph y 4 (1.4, 4.) ( 0., 1.48) 4 x (.10, 1.) c 009 La Citadelle of 1
4 . f (x + )(x 1)(x ) 7x + x By expanding, f Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry 7( x) + ( x) + 7x x f( x) = = f( x) f(x) f( x) f(x) Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = x = 1 x = y-intercept y int = f(0) = = Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f 7 + x Critical numbers are the solutions of the equation f 0 or 7 + x = 0 x = 0 ± (0) 4()( 7) x 4 = 0 (84) = 1.58 y 4 = f(x 4 ) =.8 x 5 = 0 + (84) = 1.58 y 5 = f(x 5 ) = 0.8 Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is decreasing over (, 1.58) and over (1.58, ) and is increasing over ( 1.58, 1.58). Maximum and Minimum Points The function f(x) has a minimum point at P 4 ( 1.58,.8) and a maximum point at P 5 (1.58, 0.8). Concavity Intervals The second derivative of the function f(x) is given by: f x The second derivative f (x) is zero when: f 0 x = 0 x = 0 y = f(x ) = c 009 La Citadelle 4 of 1
5 Sign Chart for the Second Derivative f (x) x f(x) f (x) + 0 Inflection Points There is an inflection point at P = (0.000, 1.500) Graph y 4 (1.5, 0.8) 4 x (0.00, 1.50) ( 1.5,.8) c 009 La Citadelle 5 of 1
6 . f (x + 1)(x 0)(x ) By expanding, f x x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = ( x) ( x) + ( x) f( x) f(x) f( x) f(x) = x x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = 1 x = 0 x = y-intercept y int = f(0) = 0 = Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f x + x Critical numbers are the solutions of the equation f 0 or x + x = 0 x = ± () 4()( ) x 4 = (8) = y 4 = f(x 4 ) = x 5 = + (8) = 1.15 y 5 = f(x 5 ) = 0.58 Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is decreasing over (, 0.549) and over (1.15, ) and is increasing over ( 0.549, 1.15). Maximum and Minimum Points The function f(x) has a minimum point at P 4 ( 0.549, 0.158) and a maximum point at P 5 (1.15, 0.58). Concavity Intervals The second derivative of the function f(x) is given by: f + x c 009 La Citadelle of 1
7 The second derivative f (x) is zero when: f 0 + x = 0 x = 1 y = f(x ) = Sign Chart for the Second Derivative f (x) x 0. f(x) f (x) + 0 Inflection Points There is an inflection point at P = (0., 0.185) Graph y 4 (1., 0.5) (0., 0.19) ( 0.55, 0.1) 4 x c 009 La Citadelle 7 of 1
8 4. f (x + )(x + )(x ) By expanding, f 1 4x + x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = 1 4( x) + ( x) + ( x) f( x) f(x) f( x) f(x) = 1 + 4x + x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = x = x = y-intercept y int = f(0) = 1 =.000 Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f + x + x Critical numbers are the solutions of the equation f 0 or + x + x = 0 x = ± ( ) 4()() x 4 = (84) =.58 y 4 = f(x 4 ) = 0.54 x 5 = + (84) = 0.58 y 5 = f(x 5 ) =.54 Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is increasing over (,.58) and over (0.58, ) and is decreasing over (.58, 0.58). Maximum and Minimum Points The function f(x) has a maximum point at P 4 (.58, 0.54) and a minimum point at P 5 (0.58,.54). Concavity Intervals The second derivative of the function f(x) is given by: f + x The second derivative f (x) is zero when: f 0 + x = 0 x = 1 y = f(x ) =.000 c 009 La Citadelle 8 of 1
9 Sign Chart for the Second Derivative f (x) x f(x).000 f (x) 0 + Inflection Points There is an inflection point at P = ( 1.000,.000) Graph y 4 (.5, 0.5) 4 x ( 1.00,.00) (0.5,.5) c 009 La Citadelle 9 of 1
10 5. f (x + 1)(x )(x ) By expanding, f + x 4x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = + ( x) 4( x) + ( x) f( x) f(x) f( x) f(x) = x 4x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = 1 x = x = y-intercept y int = f(0) = =.000 Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f 1 8x + x Critical numbers are the solutions of the equation f 0 or 1 8x + x = 0 x = 8 ± (8) 4()(1) x 4 = 8 (5) = 0.11 y 4 = f(x 4 ) =.0 x 5 = 8 + (5) =.55 y 5 = f(x 5 ) = Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is decreasing over (, 0.11) and over (.55, ) and is increasing over (0.11,.55). Maximum and Minimum Points The function f(x) has a minimum point at P 4 (0.11,.0) and a maximum point at P 5 (.55, 0.440). Concavity Intervals The second derivative of the function f(x) is given by: f 8 + x c 009 La Citadelle 10 of 1
11 The second derivative f (x) is zero when: f x = 0 x = 4 y = f(x ) = 1.9 Sign Chart for the Second Derivative f (x) x 1. f(x) 1.9 f (x) + 0 Inflection Points There is an inflection point at P = (1., 1.9) Graph y 4 (.54, 0.44) 4 x (1., 1.0) (0.1,.0) c 009 La Citadelle 11 of 1
12 . f (x + )(x + 1)(x ) By expanding, f 9 9x + x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = 9 9( x) + ( x) + ( x) f( x) f(x) f( x) f(x) = 9 + 9x + x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = x = 1 x = y-intercept y int = f(0) = 9 =.000 Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f 9 + x + x Critical numbers are the solutions of the equation f 0 or 9 + x + x = 0 x = ± ( ) 4()( 9) x 4 = (11) =.097 y 4 = f(x 4 ) = 1.8 x 5 = + (11) = 1.41 y 5 = f(x 5 ) =.4 Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is increasing over (,.097) and over (1.41, ) and is decreasing over (.097, 1.41). Maximum and Minimum Points The function f(x) has a maximum point at P 4 (.097, 1.8) and a minimum point at P 5 (1.41,.4). Concavity Intervals The second derivative of the function f(x) is given by: f + x The second derivative f (x) is zero when: f 0 + x = 0 x = 1 y = f(x ) = c 009 La Citadelle 1 of 1
13 Sign Chart for the Second Derivative f (x) x 0. f(x) f (x) 0 + Inflection Points There is an inflection point at P = ( 0., 1.975) Graph y 4 (.10, 1.8) 4 x ( 0., 1.98) (1.4,.) c 009 La Citadelle 1 of 1
14 7. f (x + )(x 0)(x 1) By expanding, f x + x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = ( x) + ( x) + ( x) f( x) f(x) f( x) f(x) = x + x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = x = 0 x = 1 y-intercept y int = f(0) = 0 = Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f + x + x Critical numbers are the solutions of the equation f 0 or + x + x = 0 x = ± ( ) 4()( ) x 4 = (8) = 1.15 y 4 = f(x 4 ) = x 5 = + (8) = y 5 = f(x 5 ) = 0.10 Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is increasing over (, 1.15) and over (0.549, ) and is decreasing over ( 1.15, 0.549). Maximum and Minimum Points The function f(x) has a maximum point at P 4 ( 1.15, 0.704) and a minimum point at P 5 (0.549, 0.10). Concavity Intervals The second derivative of the function f(x) is given by: f + x The second derivative f (x) is zero when: f 0 + x = 0 x = 1 y = f(x ) = 0.47 c 009 La Citadelle 14 of 1
15 Sign Chart for the Second Derivative f (x) x 0. f(x) 0.47 f (x) 0 + Inflection Points There is an inflection point at P = ( 0., 0.47) Graph y 4 ( 1., 0.70) ( 0., 0.5) 4 (0.55, 0.1) x c 009 La Citadelle 15 of 1
16 8. f (x + 1)(x 1)(x ) By expanding, f x x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = ( x) ( x) + ( x) f( x) f(x) f( x) f(x) = + x x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = 1 x = 1 x = y-intercept y int = f(0) = = Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f 1 x + x Critical numbers are the solutions of the equation f 0 or 1 x + x = 0 x = ± () 4()( 1) x 4 = (48) = y 4 = f(x 4 ) = 1.0 x 5 = + (48) =.155 y 5 = f(x 5 ) = 1.0 Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is increasing over (, 0.155) and over (.155, ) and is decreasing over ( 0.155,.155). Maximum and Minimum Points The function f(x) has a maximum point at P 4 ( 0.155, 1.0) and a minimum point at P 5 (.155, 1.0). Concavity Intervals The second derivative of the function f(x) is given by: f + x The second derivative f (x) is zero when: f 0 + x = 0 x = 1 y = f(x ) = c 009 La Citadelle 1 of 1
17 Sign Chart for the Second Derivative f (x) x f(x) f (x) 0 + Inflection Points There is an inflection point at P = (1.000, 0.000) Graph y 4 ( 0.15, 1.0) (1.00, 0.00) 4 x (.15, 1.0) c 009 La Citadelle 17 of 1
18 9. f (x + 1)(x )(x ) By expanding, f + x 4x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = + ( x) 4( x) + ( x) f( x) f(x) f( x) f(x) = x 4x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = 1 x = x = y-intercept y int = f(0) = = 1.00 Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f 1 8x + x Critical numbers are the solutions of the equation f 0 or 1 8x + x = 0 x = 8 ± (8) 4()(1) x 4 = 8 (5) = 0.11 y 4 = f(x 4 ) = 1.1 x 5 = 8 + (5) =.55 y 5 = f(x 5 ) = 0.17 Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is decreasing over (, 0.11) and over (.55, ) and is increasing over (0.11,.55). Maximum and Minimum Points The function f(x) has a minimum point at P 4 (0.11, 1.1) and a maximum point at P 5 (.55, 0.17). Concavity Intervals The second derivative of the function f(x) is given by: f 8 + x c 009 La Citadelle 18 of 1
19 The second derivative f (x) is zero when: f x = 0 x = 4 y = f(x ) = Sign Chart for the Second Derivative f (x) x 1. f(x) f (x) + 0 Inflection Points There is an inflection point at P = (1., 0.519) Graph y 4 (.54, 0.18) (1., 0.5) 4 x (0.1, 1.1) c 009 La Citadelle 19 of 1
20 10. f (x + )(x + )(x + 1) By expanding, f + 11x + x + x Domain The function f(x) is a polynomial function. Therefore the domain is D f = R. Symmetry f( x) = + 11( x) + ( x) + ( x) f( x) f(x) f( x) f(x) = 11x + x x Therefore the function f(x) is neither even nor odd function. Zeros The zeros of the function f(x) are: x 1 = x = x = 1 y-intercept y int = f(0) = = 1.00 Asymptotes The function f(x) is a polynomial function of degree. Therefore the function does not have any kind of asymptotes. Critical Numbers f x + x Critical numbers are the solutions of the equation f 0 or x + x = 0 x = 1 ± ( 1) 4()(11) x 4 = 1 (1) =.577 y 4 = f(x 4 ) = x 5 = 1 + (1) = 1.4 y 5 = f(x 5 ) = Sign Chart for the First Derivative f (x) x f(x) f (x) Increasing and Decreasing Intervals The function f(x) is decreasing over (,.577) and over ( 1.4, ) and is increasing over (.577, 1.4). Maximum and Minimum Points The function f(x) has a minimum point at P 4 (.577, 0.077) and a maximum point at P 5 ( 1.4, 0.077). Concavity Intervals The second derivative of the function f(x) is given by: f 1 + x The second derivative f (x) is zero when: f x = 0 x = y = f(x ) = c 009 La Citadelle 0 of 1
21 Sign Chart for the Second Derivative f (x) x.000 f(x) f (x) + 0 Inflection Points There is an inflection point at P = (.000, 0.000) Graph y 4 ( 1.4, 0.08) (.00, 0.00) (.58, 0.08) 4 x c 009 La Citadelle 1 of 1
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