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Exponential Functions 10.1

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What You Will Learn How to graph exponential functions And how to solve exponential equations and inequalities.

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Vocabulary Exponential function- an equation of the form y=ab x,where a≠0, b>0, and b≠1. Exponential growth- the base (b) is greater than 1. Exponential decay- the base is a number between 1 and 0. Exponential equation- an equation where a variable occurs as a constant. Exponential inequality- inequalities involving exponential functions.

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Characteristics of Exponential Functions The function is continuous and one-to-one The domain is the set of all real numbers. The x-axis is an asymptote of the graph. The range is the set of all positive numbers if a>0 and all negative if a<0. The graph contains the point (0,a). That is, the y- intercept is a. The graphs of y=ab x and y=a(1/b) x are reflections across the y-axis.

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Types of Exponential Functions Exponential growth- – If a>0 and b>1, the function y=ab x represents exponential growth. Exponential Decay- – If a>0 and 0<b<1, the function y=ab x represents exponential decay.

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Determine Growth or Decay y=2(6) x – This function represents exponential growth, since the base is greater than 1. y=(1/5) x – This function, however, represents exponential decay since the base is between 1 and 0.

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Graphing and Exponential Function x Y=3(1/2) x -3 3(1/2) -3 =24 -23(1/2) -2 =12 3(1/2) -1 =6 03(1/2) 0 =3 13(1/2) 1 =1.5 23(1/2) 2 =0.75 33(1/2) 3 =.375 The domain of the function is all real numbers, while the range is the set of all positive numbers Sketch the graph of y=3(1/2) x. Then state the function’s domain and range.

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Exponential Equations If b is a positive number other than 1, then b x =b y. If and only if (IFF) x=y. If 5 x =5 7, then x=7. Example: – 3 2n+1 =81 – 3 2n+1 =3 4 Rewrite 8 so that each base is the same – 2n+1=4 Property of Equality for Exponential Functions – 2n=3 – n=3/2

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Exponential Inequalities 4 3p-1 > 1/256 – 4 3p-1 > 4 -4 find a common base – 3p-1 > -4 Property of inequality for Exponential Functions – 3p > -3 – p > -1

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Simplify expressions with irrational exponents 2 √5 X 2 √3 Add the exponents to get… 2 √5 + √3

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