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Algebra II Chapter 5 Notes Day 1: 5.1 Use Properties of Exponents 1. Product of powers: When the bases are the_____– add the exponents Ex.1: 53 51 531 52 25 2. Power of a power: When have one base with ____ powers - _________ the exponents. Ex.2: (32 )3 32 3 36 729 3. Power of a product: When two bases with _____ exponent - ______ the bases and raise to power of the exponent. Ex.3: 24 34 (2 3)4 64 1296 4. Negative exponent: When have a base with a ________ exponent- _____ the base and raise to positive power. DO NOT make the answer negative. 2 1 1 Ex.4: 7 2 49 7 5. Zero exponent: Anything raised to the _____ power is 1. Ex.5: (89)0 1 6. Quotient of powers: When the bases are the same in a division problem -_______ the exponents. 63 1 Ex.6: 2 63( 2) 63 2 61 6 6 Try these: 1. 2 3 2. 4 4 2 3 2 4. 3 4 6 7 5. b b b 3 r 2 3. 3 = s 16m4 n 5 6. 2m1n 5 Scientific Notation: Use properties of exponents Ex. 1: 8.5 X 107 1.2 X 103 Ex. 2: 1.1X 103 5.5 X 108 1 Day 26:5.2 Evaluate and graph polynomial functions Polynomial function: f ( x) ax n bx n1 cx n2 ... mx1 n Where: a 0 and the exponents are all _____ numbers, and the coefficients are all _____ numbers. Leading coefficient: The coefficient of the term with the highest exponent. Degree: The degree is the value of the highest exponent. Type Constant Linear Quadratic Common Polynomial Functions Standard Form f ( x) a f ( x) ax b Example f ( x) 14 f ( x) 5 x 7 f ( x) ax 2 bx1 c f ( x) 2 x 2 x 9 3 Cubic f ( x) ax3 bx3 cx1 d f ( x) x 3 x 2 3x 4 Quartic f ( x) ax 4 bx3 cx 2 dx1 e f ( x) x 4 2 x 1 Degree 0 1 2 Decide whether the function is a polynomial function. If so write in standard form, state its degree, type, and leading coefficient. 1 1. h( x) x 4 x 2 3 2. g ( x) 7 x 3 x 2 4 3. f ( x) 5x 2 3x 1 x 4. k ( x) x 2x 0.6 x5 Evaluate by direct substitution: Plug in value given for x into function wherever x is. Use direct substitution to evaluate f ( x) 2 x 4 5 x3 4 x 8 when x=3. f (3) 2(3)4 5(3)3 4(3) 8 = Try: #1: f ( x) x 4 2 x3 3x 2 7; x 2 #2: g ( x) x3 5x 2 6 x 1; x 4 2 Synthetic substitution: How to evaluate a function more easily than direct substitution Use synthetic substitution to evaluate f ( x) 2 x 4 5 x3 4 x 8 when x=3. Step 1: Write coefficients in descending order using zeros for terms missing. 3 2 -5 0 -4 8 Step 2: Bring down first coefficient. Multiply that by x-value. Write the product under the second coefficient. Repeat until done. Step 3: Final sum is value of f(x) when x = 3. Try these: #1: f ( x) x 4 2 x3 3x 2 7; x 2 #2: g ( x) x3 5x 2 6 x 1; x 4 End Behavior : Behavior of the graph of a function as x approaches positive infinity ( , ) or negative infinity ( , ) .It is determined by the function’s ________ and the sign of the ___________ coefficient. End Behavior of Graphs of Polynomial Functions Odd-degree polynomial functions have graphs with opposite behavior at each end. Even-degree polynomial functions have graphs with the same behavior at each end. The Leading Coefficient Test The coefficient on the term with the largest exponent on the variable is called the leading coefficient. If the largest exponent is an odd number: If the leading coefficient is positive, the graph falls to the left and rises to the right. Rises Left If the leading coefficient is negative, the graph rises to the left and falls to the right. Rises Right Falls Left Falls Right Odd Degree; positive leading coefficient Odd Degree; negative leading coefficient 3 The Leading Coefficient Test (continued) The coefficient on the term with the largest exponent on the variable is called the leading coefficient. If the largest exponent is an even number: If the leading coefficient is positive, the graph rises to the left and rises to the right. If the leading coefficient is negative, the graph falls to the left and falls to the right. Falls Right Rises Right Rises Left Falls Left Even Degree; negative leading coefficient Even Degree; positive leading coefficient Graphing polynomial functions: Make a table of values and plot the points. Connect the points into a smooth curve and check end behaviors. Graph f ( x) x3 x 2 3x 3 x -3 -2 -1 0 1 2 y 3 4