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Evaluate and Graph Polynomial Functions
Section 5.2 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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Common Polynomial Functions
POLYNOMIAL is an expression made with constants, variables and exponents, which are combined using an operation. The exponents can only be WHOLE numbers. Another word for exponents is a degree. Leading Coefficient is the number in front of the highest polynomial Types of coefficient: First Degree polynomial is a LINEAR function Second Degree polynomial is a QUADRATIC function Third Degree polynomial is a CUBIC function Fourth Degree polynomial is a QUARTIC function Fifth Degree polynomials and beyond are known by their degree functions. 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 1 Is 4x3 + 6x4 + 5x2 – 7x + 10 a polynomial? If so, write it in standard form and identify its degree, type, leading coefficient, and constant term. 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 2 Is a polynomial? If so, write it in standard form and identify its degree, type, leading coefficient, and constant term. 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 3 Is 3 + x3 – 1/4x2 a polynomial? If so, write it in standard form and identify its degree, type, leading coefficient, and constant term. 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Your Turn Is a polynomial? If so, write it in standard form and identify its degree, type, leading coefficient, and constant term. 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
End Behavior FIRST DEGREE polynomial is a straight-line SECOND DEGREE polynomial is a parabola Every graph is continuous. There are no gaps, jumps, holes, or sharp corners. End Behavior: When a polynomial function has an ODD degree, one end of its graph shoots upward and the other end downwards. When a polynomial function has an EVEN degree, both ends will shoot upwards or downwards The leading coefficient focuses on the RIGHT side. If it is POSITIVE, the graph will RISE TO THE RIGHT If it is NEGATIVE, the graph will FALL TO THE RIGHT The highest degree focuses on the LEFT side. EVEN: The left behavior is the SAME as the right behavior; EQUAL EVEN ODD: The left behavior is the OPPOSITE of the right behavior; OPPOSITE ODD 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions 7
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5.2: Evaluate and Graph Polynomial Functions
Example 4 How would you describe the graph from negative x to positive x? 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 5 How would you describe the graph from negative x to positive x? 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Your Turn How would you describe the graph from negative x to positive x? 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
End Behavior The leading coefficient focuses on the RIGHT side. If it is POSITIVE, the graph will RISE TO THE RIGHT If it is NEGATIVE, the graph will FALL TO THE RIGHT The highest degree focuses on the LEFT side. EVEN: The left behavior is the SAME as the right behavior; EQUAL EVEN ODD: The left behavior is the OPPOSITE of the right behavior; OPPOSITE ODD 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions 11
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5.2: Evaluate and Graph Polynomial Functions
Example 6 Identify the end behavior of f(x) = 2x5 + 3x2 – 4x – 1 What is the leading coefficient? Is it positive or negative? As x +∞, f(x) +∞ 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 6 Identify the end behavior of f(x) = 2x5 + 3x2 – 4x – 1 What is the highest degree? Is it odd or even? As x +∞, f(x) +∞ As x –∞, f(x) –∞ 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 6 Identify the end behavior of f(x) = 2x5 + 3x2 – 4x – 1 As x –∞, f(x) –∞ As x +∞, f(x) +∞ 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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Example 7 Identify the end behavior of f(x) = –0.6x5 + 4x2 + x – 4
As x –∞, f(x) +∞ As x +∞, f(x) –∞ 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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Example 8 Identify the end behavior of f(x) = x4 – 3x3 + 5x
As x –∞, f(x) +∞ As x +∞, f(x) +∞ 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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Your Turn Identify the end behavior of f(x) = –2x4 + 8x3 – 8x2
As x –∞, f(x) –∞ As x +∞, f(x) –∞ 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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Steps in Identifying Graphs
Plug in every x with a negative x and solve EVEN FUNCTION Even Functions are functions that are symmetric to the y–axis. When substituting x, the original equation is EQUAL to the new equation C. ODD FUNCTION Odd Functions are functions that cross the origin. When substituting x, the original equation is OPPOSITE to the new equation For an odd function, rotating the graph through 180 degrees would give the same graph. D. NEITHER Functions that is not symmetric on the y–axis and at the origin. OPPOSITE ODD EQUAL EVEN 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 9 Given f(x) = x2 – 5, is the function even, odd, or neither? EVEN FUNCTION 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 9 Given f(x) = x2 – 5, is the function even, odd, or neither? EVEN FUNCTION 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 10 Given f(x) = –5x3 + x, is the function even, odd, or neither? ODD FUNCTION 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 10 Given f(x) = –5x3 + x, is the function even, odd, or neither? ODD FUNCTION 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 11 Given f(x) = x3 – 1, is the function even, odd, or neither? NEITHER 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 11 Given f(x) = x3 – 1, is the function even, odd, or neither? NEITHER 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Your Turn Given f(x) = 2x3 – 3x2 – 4x + 4, is the function even, odd, or neither? NEITHER 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 12 Is this graph an odd function, even function, or neither? ODD FUNCTION 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 13 Is this graph an odd function, even function, or neither? EVEN FUNCTION 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Example 14 Is this graph an odd function, even function, or neither? NEITHER 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Your Turn Is this graph an odd function, even function, or neither? ODD FUNCTION 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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5.2: Evaluate and Graph Polynomial Functions
Assignment Worksheet 11/6/2018 6:25 PM 5.2: Evaluate and Graph Polynomial Functions
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