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5.2 Evaluating and Graphing Polynomial Functions DAY 1
Goal: Evaluate and graph polynomial functions.
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n n – 1 f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0
EVALUATING POLYNOMIAL FUNCTIONS A polynomial function is a function of the form f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0 constant term an 0 an leading coefficient descending order of exponents from left to right. n n – 1 n degree Where an 0 and the exponents are all whole numbers. For this polynomial function, an is the leading coefficient, a 0 is the constant term, and n is the degree. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.
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Common Types of Polynomials
Degree Type Standard Form Constant f(x) = a (f(x) = -5) 1 Linear f(x) = x (f(x) = 3x) 2 Quadratic f(x) = x2 + x + a0 3 Cubic f(x) = x3 + x2 + x + a0 4 Quartic f(x) = x4 + x3 + x2 + x + a0
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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.
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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.
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Evaluating a Polynomial Function
One way to evaluate is to use direct substitution. PLUG IT IN, PLUG IT IN
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Using Synthetic Substitution
One way to evaluate polynomial functions is to use direct substitution. Another way to evaluate a polynomial is to use synthetic substitution. Use synthetic division to evaluate f (x) = 2 x x x - 7 when x = 3.
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Using Synthetic Substitution
SOLUTION 2 x x 3 + (–8 x 2) + 5 x + (–7) Polynomial in standard form Polynomial in standard form 2 0 –8 5 –7 3 x-value 3 • Coefficients Coefficients 6 18 30 105 2 6 10 35 98 THE REMAINDER The value of f (3) is the last number you write, In the bottom right-hand corner.
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Evaluate using Direct Substitution and Synthetic Substitution
Direct Substitution Synthetic Substitution
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Evaluate the Polynomial Function Using Direct Substitution when x = -2
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Evaluate the Polynomial Function Using Synthetic Substitution
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POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1 Max. Zeros: 1
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POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2 Max. Zeros: 2
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POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3 Max. Zeros: 3
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POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function Degree = 4 Max. Zeros: 4
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f(x) = x3 POLYNOMIAL FUNCTIONS Degree: Odd Leading Coefficient: +
END BEHAVIOR f(x) = x3 Degree: Odd Leading Coefficient: + End Behavior: As x -∞; f(x) -∞ As x +∞; f(x) +∞
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POLYNOMIAL FUNCTIONS f(x) = -x3 Degree: Odd Leading Coefficient: –
END BEHAVIOR f(x) = -x3 Degree: Odd Leading Coefficient: – End Behavior: As x -∞; f(x) +∞ As x +∞; f(x) -∞
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POLYNOMIAL FUNCTIONS f(x) = x4 Degree: Even Leading Coefficient: +
END BEHAVIOR f(x) = x4 Degree: Even Leading Coefficient: + End Behavior: As x -∞; f(x) +∞ As x +∞; f(x) +∞
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POLYNOMIAL FUNCTIONS f(x) = -x4 Degree: Even Leading Coefficient: –
END BEHAVIOR f(x) = -x4 Degree: Even Leading Coefficient: – End Behavior: As x -∞; f(x) -∞ As x +∞; f(x) -∞
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End behavior – four cases
Graphing Summary Degree -- highest power exponent or # of factors with variable terms Number of turns -- if there are n zeros, then there are n-1 turns Leading coefficient– coefficient of highest power variable or the number in front of all the factors X-intercepts –the zeros, found by setting each factor equal to zero Y- intercept – where the graph crosses the y axis, found by evaluating the function for an x-value of 0 End behavior – four cases Even degree, + leading coefficient –both ends point up Even degree, - leading coefficient—both ends point down Odd degree, + leading coefficient—starts down, ends up Odd degree, - leading coefficient—starts up, ends down
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5.2 Continued: Graphing Polynomial Functions
Will use end behavior to analyze the graphs of polynomial functions.
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End Behavior Behavior of the graph as x approaches positive infinity (+∞) or negative infinity (-∞) The expression x→+∞ : as x approaches positive infinity The expression x→-∞ : as x approaches negative infinity
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End Behavior of Graphs of Linear Equations
f(x) = x f(x) = -x f(x)→+∞ as x→+∞ f(x)→-∞ as x→-∞ f(x)→-∞ as x→+∞ f(x)→+∞ as x→-∞
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End Behavior of Graphs of Quadratic Equations
f(x) = x² f(x) = -x² f(x)→+∞ as x→+∞ f(x)→+∞ as x→-∞ f(x)→-∞ as x→+∞ f(x)→-∞ as x→-∞
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Investigating Graphs of Polynomial Functions
How does the sign of the leading coefficient affect the behavior of the polynomial function graph as x→+∞? How is the behavior of a polynomial functions graph as x→+∞ related to its behavior as x→-∞ when the functions degree is odd? When it is even?
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Graphing Polynomial Functions
f(x)= x³ + x² – 4x – 1 x -3 -2 -1 1 2 3 f(x)
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Graphing Polynomial Functions
f(x)= -x4 – 2x³ + 2x² + 4x x -3 -2 -1 1 2 3 f(x)
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