Standard form: terms are written in descending order of exponents from left to right. Leading Coefficient: the coefficient of the variable with the highest.

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Presentation transcript:

standard form: terms are written in descending order of exponents from left to right. Leading Coefficient: the coefficient of the variable with the highest degree Degree: the largest exponent on a variable 6.2 Evaluating & Graphing Polynomial Functions Do Now: What is the coefficient? -6x + 5 OBJ: To Evaluate a polynomial function & Graph a polynomial function

DegreeTypeStandard Form common types of polynomial functions: 4Quartic f (x) = 4x 4 + x 3 - 9x 2 + x Constantf (x) = 3 3Cubic f (x) = x 3 - 4x x - 8 2Quadratic f (x) = -2 x x + 1 1Linearf (x) = 5x - 2

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = x 2 – 3x 4 – polynomial function: variable base with whole # exponents degree: 4, so it is a quartic function leading coefficient: – 3 standard form: f(x)=-3x 4 + ½ x )

NOT a polynomial function: because a term does not have a variable base with an exponent that is a whole number. f (x) = x x 2) f (x) = 6x x – 1 + x NOT a polynomial function: exponent that is not a whole number. 3)

f (x) = – 0.5 x +  x 2 – 2 Polynomial Function Degree: 2, so it is a quadratic function. Leading coefficient:  standard form: f (x) =  x 2 – 0.5x – 2. 4) Polynomial function? (DO NOT COPY) f (x) = x 2 – 3 x 4 – f (x) = x x f (x) = 6x x – 1 + x f (x) = – 0.5x +  x 2 – 2

Use synthetic substitution (aka synthetic division) to evaluate: f (x) = 2 x 4 +  8 x x  7 when x = 3. Using Synthetic Substitution 5)

Polynomial in standard form 2 x x 3 + (–8 x 2 ) + 5 x + (–7) The value of f (3) is the last number you write, In the bottom right-hand corner. The value of f (3) is the last number you write, In the bottom right-hand corner. 20–85 –720–85 –7 Coefficients 3 x -value 3 S OLUTION Polynomial in standard form

Homework

End Behavior is what happens to f(x) as x gets very large (+  ) or very small (-  ) it depends on the degree (odd or even) and the leading coefficient (positive or negative) *The expression x +  is read as “x approaches positive infinity.” OBJ: describe the end behavior of a polynomial function DO NOW: complete handout

END BEHAVIOR FOR POLYNOMIAL FUNCTIONS Even degree: With positive LC: as x + , f(x) +  as x - , f(x) +  With negative LC: as x + , f(x) -  as x - , f(x) -  Odd degree: With positive LC: as x + , f(x) +  as x - , f(x) -  With negative LC: as x + , f(x) -  as x - , f(x) + 

1) f (x) = x 3 + x 2 – 4 x – 1 The degree is ODD and the LC is POSITIVE so, as x + , f(x) +  and as x - , f(x) -  Examples: 2) f (x) = –x 4 – 2x 3 + 2x 2 + 4x The degree is EVEN and the LC is NEGATIVE so, as x + , f(x) -  and as x - , f(x) - 

HW:

x f (x) –3 –7 –2 3 – – S OLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. 1) f (x) = x 3 + x 2 – 4 x – 1 Graphing Polynomial Functions

x f (x) –3 –21 –2 0 – –16 3 –105 2) Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x S OLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.