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**Exponential Functions**

Chapter 4 Exponential Functions

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**4.1 Properties of Exponents**

Know the meaning of exponent, zero exponent and negative exponent. Know the properties of exponents. Simplify expressions involving exponents Know the meaning of exponential function. Use scientific notation.

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**Exponent For any counting number n,**

We refer to as the power, the nth power of b, or b raised to the nth power. We call b the base and n the exponent.

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**Examples When taking a power of a negative number,**

if the exponent is even the answer will be positive if the exponent is odd the answer will be negative

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**Properties of Exponents**

Product property of exponents Quotient property of exponents Raising a product to a power Raising a quotient to a power Raising a power to a power

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**Meaning of the Properties**

Product property of exponents Raising a quotient to a power

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**Simplifying Expressions with Exponents**

An expression is simplified if: It included no parenthesis All similar bases are combined All numerical expressions are calculated All numerical fractions are simplified All exponents are positive

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**Order of Operations Parenthesis Exponents Multiplication Division**

Addition Subtraction

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Warning Note: When using a calculator to equate powers of negative numbers always put the negative number in parenthesis. Note: Always be careful with parenthesis

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Examples

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Examples (Cont.)

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Zero Exponent For b ≠ 0, Examples,

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**Negative Exponent If b ≠ 0 and n is a counting number, then**

To find , take its reciprocal and switch the sign of the exponent Examples,

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**Negative Exponent (Denominator)**

If b ≠ 0 and n is a counting number, then To find , take its reciprocal and switch the sign of the exponent Examples,

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**Simplifying Negative Exponents**

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**Exponential Functions**

An exponential function is a function whose equation can be put into the form: Where a ≠ 0, b > 0, and b ≠ 1. The constant b is called the base.

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**Exponential vs Linear Functions**

x is a exponent x is a base

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**Scientific Notation A number written in the form:**

where k is an integer and -10 < N ≤ -1 or 1 ≤ N < 10 Examples

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**Scientific to Standard Notation**

When k is positive move the decimal to the right When k is negative move the decimal to the left move the decimal 3 places to the right move the decimal 5 places to the left

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**Standard to Scientific Notation**

if you move the decimal to the right, then k is positive if you move the decimal to the left, then k is negative move the decimal 4 places to the left move the decimal 9 places to the right

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Group Exploration If time, p173

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4.2 Rational Exponents

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**Rational Exponents ( ) For the counting number n, where n ≠ 1,**

If n is odd, then is the number whose nth power is b, and we call the nth root of b If n is even and b ≥ 0, then is the nonnegative number whose nth power is b, and we call the principal nth root of b. If n is even and b < 0, then is not a real number. may be represented as

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**Examples ½ power = square root ⅓ power = cube root**

not a real number since the 4th power of any real number is non-negative

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Rational Exponents For the counting numbers m and n, where n ≠ 1 and b is any real number for which is a real number, A power of the form or is said to have a rational exponent.

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Examples

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**Properties of Rational Exponents**

Product property of exponents Quotient property of exponents Raising a product to a power Raising a quotient to a power Raising a power to a power

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Examples

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**4.3 Graphing Exponential Functions**

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**Graphing Exponential Functions by hand**

-3 1/8 -2 1/4 -1 1/2 1 2 4 3 8

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**Graph of an exponential function is called an exponential curve**

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x y -1 8 4 1 2 3 1/2

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**Base Multiplier Property**

For an exponential function of the form If the value of the independent variable increases by 1, then the value of the dependent variable is multiplied by b.

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**x increases by 1, y increases by b**

-3 1/8 -2 1/4 -1 1/2 1 2 4 3 8 x y -1 8 4 1 2 3 1/2

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**Increasing or Decreasing Property**

Let , where a > 0. If b > 1, then the function is increasing grows exponentially If 0 < b < 1, then the function is decreasing decays exponentially

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Intercepts y-intercept for the form: is (0,a) is (0,1)

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**Intercepts Find the x and y intercepts: y-intercept x-intercept**

as x increases by 1, y is multiplied by 1/3. infinitely multiplying by 1/3 will never equal 0 as x increases, y approaches but never equals 0 no x-intercept exists, instead the x-axis is called the horizontal asymptote

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**Reflection Property The graphs**

are reflections of each other across the x-axis a > 0 a > 0 a < 0 a < 0

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exponential functions

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