Section 10-1. Vocabulary: Exponential function- In general, an equation of the form, where, b>0, and, is known as an exponential function. Exponential.

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Presentation transcript:

Section 10-1

Vocabulary: Exponential function- In general, an equation of the form, where, b>0, and, is known as an exponential function. Exponential growth- The base of an exponential growth function is greater than 1. In an exponential function, the base is a constant and the exponent is a variable. Exponential decay- The base of an exponential decay function is a number between 0 and 1. Exponential equations- Equations in which variables occur as exponents.

Graph of an exponential function To sketch the graph of, make a table of values and connect them to make a smooth curve. XY -31/8 -21/4 ½ As the value of x decreases, the value of y approaches zero.

Characteristics of Exponential Functions: ♠T♠The function is continuous and one-to-one. ♠ The domain is the set of all real numbers. he x-axis is an asymptote of the graph. he range is the set of all positive numbers if a>0 and all negative numbers if a<0. he graph contains the point (0,a), so therefore, a is the y-intercept. he graphs of and are reflections over the y-axis.

Exponential growth and decay If a>0, and b>1, the function represents exponential growth. (the value of x grows/rises from left to right). Exponential growth Exponential decay If a>0, and 0<b<1, then the function represents exponential decay.

Expressions with Irrational Exponents Note: Since the domain of an exponential function includes irrational numbers, the properties of rational exponents apply to irrational exponents. Simplify each expression: 1. 2.

Exponential Equations Property of Equality for Exponential Functions: ☻I☻If b is a positive number other than 1, then, if and only if x=y. ☻E☻Example: if, then x=8. Solve each equation: 1. (Rewrite 81 in base 3 so both sides are in base 3) 2.

Exponential Inequalities Property of Inequality for Exponential Functions: ¤ If b>1, then if and only if x>y, and if and only if x<y. ¤ Example: if, then x<8. Solve the following equation:

Correlation to Reflexes (very complicated)

Homework Page 529 #21-49 odd