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Exponential Functions Chapter 4

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

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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 9 places to the right move the decimal 4 places to the left

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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 4 th 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 xy -31/8 -21/4 1/2 01 12 24 38

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Graph of an exponential function is called an exponential curve

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xy 8 04 12 21 31/2

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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 xy -31/8 -21/4 1/2 01 12 24 38 xy 8 04 12 21 31/2

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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) y-intercept for the form: 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

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