Exponential Functions Chapter 4 Exponential Functions

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.

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.

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

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

Meaning of the Properties Product property of exponents Raising a quotient to a power

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

Order of Operations Parenthesis Exponents Multiplication Division Addition Subtraction

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

Examples

Examples (Cont.)

Zero Exponent For b ≠ 0, Examples,

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,

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,

Simplifying Negative Exponents

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.

Exponential vs Linear Functions x is a exponent x is a base

Scientific Notation A number written in the form: where k is an integer and -10 < N ≤ -1 or 1 ≤ N < 10 Examples

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

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

Group Exploration If time, p173

4.2 Rational Exponents

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 .

Examples ½ power = square root ⅓ power = cube root not a real number since the 4th power of any real number is non-negative

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.

Examples

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

Examples

4.3 Graphing Exponential Functions

Graphing Exponential Functions by hand -3 1/8 -2 1/4 -1 1/2 1 2 4 3 8

Graph of an exponential function is called an exponential curve

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

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.

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

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

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

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

Reflection Property The graphs are reflections of each other across the x-axis a > 0 a > 0 a < 0 a < 0