8.2 Negative and Zero Exponents I love exponents!

43210 In addition to level 3, students make connections to other content areas and/or contextual situations outside of math. Students will construct, compare, and interpret linear and exponential function models and solve problems in context with each model. - Compare properties of 2 functions in different ways (algebraically, graphically, numerically in tables, verbal descriptions) - Describe whether a contextual situation has a linear pattern of change or an exponential pattern of change. Write an equation to model it. - Prove that linear functions change at the same rate over time. - Prove that exponential functions change by equal factors over time. - Describe growth or decay situations. - Use properties of exponents to simplify expressions. Students will construct, compare, and interpret linear function models and solve problems in context with the model. - Describe a situation where one quantity changes at a constant rate per unit interval as compared to another. Students will have partial success at a 2 or 3, with help. Even with help, the student is not successful at the learning goal. Focus 8 Learning Goal – (HS.N-RN.A.1 & 2, HS.A-SSE.B.3, HS.A-CED.A.2, HS.F-IF.B.4, HS.F- IF.C.8 & 9, and HS.F-LE.A.1) = Students will construct, compare and interpret linear and exponential function models and solve problems in context with each model.

Definition of Negative Exponents (let a be a nonzero number and let n be a positive integer) The expression a -n is the reciprocal of a n. a -n = 1 a ≠ 0 a n 1 = a n a -n 3 -2 = = 1 9

The negative exponent says the number needs to be moved to the opposite location and made positive. If it’s negative in the numerator, it belongs in the denominator position positive. If it’s negative in the denominator position, it belongs in the numerator position positive.

Definition of Zero Exponent (let a be a nonzero number and let n be a positive integer) A nonzero number to the zero power is ALWAYS 1! a 0 = 1a  = 1 (x 2 y 5 ) 0 = 1 The expression 0 0 is undefined.

Simplify expressions: write with positive exponents. (-5) -3 = 24  4 -3 = = 1 (-5) 3 = ∙ = 24∙ 1 = = -

Simplify expressions: write with positive exponents. 3a -3 b -2 = 3 a3b2a3b2 (3 -3 ) 2 =3 (-3 ∙ 2) = 3 -6 = = 1 729

Graphing with a variable as an exponent x x2x =12 -1 = ½2 -2 = ¼2 -3 = 1 / 8 Sketch the graph of y = 2 x Will it ever touch the x-axis?

Example: (Just follow this example to see what you get to do on your assignment.) Between 1970 and 1990, the Missouri population increased at a rate of.47% per year. The population P in t years is given by: P = 4,903,000  t Where t = 0 for 1980

To find the population, plug the numbers into the formula and then use a calculator. Find the population in 1970, 1980, and 1990 Pop in 1970t = -10 Pop in 1990t = 10 P = 4,903,000  (set up the problem) = 4,678,406(calculate) This is the population in Do you expect it to be more or less in 1970 than 1980?

Population in 1990: P = 4,903,000  P = 5,138,376