# polynomial functions mct4c1. polynomial functions the largest exponent within the polynomial...

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Polynomial FunctionsMCT4C1

Polynomial FunctionsThe largest exponent within the polynomial determines the degree of the polynomial.

Polynomial Function in General FormDegreeName of Function1Linear2Quadratic3Cubic4Quartic

Symmetry in Polynomial FunctionsLine symmetry must reflect across y-axis. Rotational symmetry must rotate 1800 about origin.

Explore PolynomialsLinear FunctionQuadratic FunctionCubic FunctionQuartic Function

Leading CoefficientThe leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.For example, the quartic function f(x) = -2x4 + x3 5x2 10 has a leading coefficient of -2.

Cubic PolynomialsLook at the two graphs and discuss the questions given below.1. How can you check to see if both graphs are functions?3. What is the end behaviour for each graph?4. Which graph do you think has a positive leading coeffient? Why?5. Which graph do you think has a negative leading coefficient? Why?2. How many x-intercepts do graphs A & B have?

Cubic PolynomialsThe following chart shows the properties of the graphs on the left.

EquationFactored form & Standard formX-InterceptsSign of Leading CoefficientEnd BehaviourDomain and RangeFactoredy=(x+1)(x+4)(x-2)Standardy=x3+3x2-6x-8-4, -1, 2PositiveAs x, y and x-, y- Domain {x| x R}Range{y| y R}

Factoredy=-(x+1)(x+4)(x-2)Standardy=-x3-3x2+6x+8-4, -1, 2NegativeAs x, y- and x-, yDomain{x| x R}Range{y| y R}

Cubic PolynomialsThe following chart shows the properties of the graphs on the left.

EquationFactored form & Standard formX-InterceptsSign of Leading CoefficientEnd BehaviourDomain and RangeFactoredy=(x+3)2(x-1)Standardy=x3+5x2+3x-9-3, 1PositiveAs x, y and x-, y- Domain {x| x R}Range{y| y R}

Factoredy=-(x+3)2(x-1)Standardy=-x3-5x2-3x+9-3, 1NegativeAs x, y- and x-, yDomain{x| x R}Range{y| y R}

Cubic PolynomialsThe following chart shows the properties of the graphs on the left.

EquationFactored form & Standard formX-InterceptsSign of Leading CoefficientEnd BehaviourDomain and RangeFactoredy=(x-2)3Standardy=x3-6x2+12x-82PositiveAs x, y and x-, y-Domain {x| x R}Range{y| y R}

Factoredy=-(x-2)3Standardy=-x3+6x2-12x+82NegativeAs x, y- and x-, yDomain{x| x R}Range{y| y R}

Quartic PolynomialsLook at the two graphs and discuss the questions given below.1. How can you check to see if both graphs are functions?3. What is the end behaviour for each graph?4. Which graph do you think has a positive leading coeffient? Why?5. Which graph do you think has a negative leading coefficient? Why?2. How many x-intercepts do graphs A & B have?

Quartic PolynomialsThe following chart shows the properties of the graphs on the left.

EquationFactored form & Standard formX-InterceptsSign of Leading CoefficientEnd BehaviourDomain and RangeFactoredy=(x-3)(x-1)(x+1)(x+2)Standardy=x4-x3-7x2+x+6-2,-1,1,3PositiveAs x, y and x-, yDomain {x| x R}Range{y| y R, y -12.95}

Factoredy=-(x-3)(x-1)(x+1)(x+2)Standardy=-x4+x3+7x2-x-6-2,-1,1,3NegativeAs x, y- and x-, y-Domain{x| x R}Range{y| y R,y 12.95}

Quartic PolynomialsThe following chart shows the properties of the graphs on the left.

EquationFactored form & Standard formX-InterceptsSign of Leading CoefficientEnd BehaviourDomain and RangeFactoredy=(x-4)2(x-1)(x+1)Standardy=x4-8x3+15x2+8x-16-1,1,4PositiveAs x, y and x-, yDomain {x| x R}Range{y| y R,y -16.95}

Factoredy=-(x-4)2(x-1)(x+1)Standardy=-x4+8x3-15x2-8x+16-1,1,4NegativeAs x, y- and x-, y-Domain{x| x R}Range{y| y R,y 16.95}

Quartic PolynomialsThe following chart shows the properties of the graphs on the left.

EquationFactored form & Standard formX-InterceptsSign of Leading CoefficientEnd BehaviourDomain and RangeFactoredy=(x+2)3(x-1)Standardy=x4+5x3+6x2-4x-8-2,1PositiveAs x, y and x-, yDomain {x| x R}Range{y| y R, y -8.54}

Factoredy=-(x+2)3(x-1)Standardy=-x4-5x3-6x2+4x+8-2,1NegativeAs x, y- and x-, y-Domain{x| x R}Range{y| y R,y 8.54}

Quartic PolynomialsThe following chart shows the properties of the graphs on the left.

EquationFactored form & Standard formX-InterceptsSign of Leading CoefficientEnd BehaviourDomain and RangeFactoredy=(x-3)4Standardy=x4-12x3+54x2-108x+813PositiveAs x, y and x-, yDomain {x| x R}Range{y| y R, y 0}

Factoredy=-(x-3)4Standardy=-x4+12x3-54x2+108x-813NegativeAs x, y- and x-, y-Domain{x| x R}Range{y| y R,y 0}

Common Differences1842669011424242424Since it is a 3rd CommonDifference, the function is CUBIC.

The leading coefficient isPositive.

The leading coefficient canbe found using: 24 = a(3!) 24 = 6a 4 = a

*Teachers: This definition for degree has been simplified intentionally to help students understand the concept quickly and easily.*Teachers: This definition for degree has been simplified intentionally to help students understand the concept quickly and easily.

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