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

4.4 Finding Equations of Exponential Functions

Finding an Equation Using a Table Refer to the Base Multiplier Property as x increases by 1, y is multiplied by the base Find the y-intercept (0, a) = (0, 2) Find the constant multiplying by 4 Write the equation x F(x) 2 1 8 32 3 148 4

Linear vs Exponential x F(x) 243 1 81 2 27 3 9 4 x F(x) 81 1 70 2 59 3 243 1 81 2 27 3 9 4 x F(x) 81 1 70 2 59 3 48 4 37

Solving for b No Real Solutions

When Solving For b For , For , n is even, always have a positive and negative answer n is odd, always have one positive answer For , n is even, always no real solutions n is odd, always have one negative answer

Solving for b (cont) Remember to always simplify both sides of the equations first.

Finding an Equation of Exponential Curves Using Two Points Given (0,4) and (5,128) We know (0,4) is y-intercept (0,a) so a = 4 Substitute (5, 128) into the equation

Example (2,1) (5,7) Plug both points into the standard equation Divide to cancel a term

Example (cont) Now we have b We can substitute one of the points and solve for a …..(2,1)

4.5 Using Exponential Functions to Model Data

Exponential Models Exponential model – exponential function, or its graph, that describes the relationship between two quantities in an authentic situation Exponentially related – If all the data points for a situation lie on an exponential curve Approximately exponentially related – If all the data points lie on or close to an exponential curve

Suppose there are 5 million bacteria on a banana at 8am on Monday Suppose there are 5 million bacteria on a banana at 8am on Monday. Every bacterium divides into 2 every hour, on average. Give a function for the situation Predict the number of bacteria at 8pm on Monday. 20,480 million bacteria

Exponential Functions of Time y is the amount t amount of time a is the initial amount when t =0 b is constant by which a grows or decays over time

Write an equation to model the data A person invests 8,000 in an account and interest is compounded 5% annually. Write an equation to model the data Value is 100% of the original deposit plus 6% after the first year so we use 1.06 as our base What is the value after 10 years? The value would be $14,326.78 after 10 years

Half-life If a quantity decays exponentially, the half-life is the amount of time it takes for that quantity to be reduced to half half life

Californium-251 is a radioactive element with a half-life of 900 years. Give an equation to model the data. What percent would be left after 600 years? about 63%

Base of Exponential Model b > 1, grows exponentially by a rate of b – 1 0 < b < 1, decays exponentially by a rate of 1 – b Notice: b ≠ 1, always equals one therefore the equation would be a horizontal line at the initial amount (a)

A person’s heart attack risk can be estimated by using Framingham point scores. If men’s risk of a heart attack is 1% at a Fram-score of 0, and 20% at a Fram-score of 15. Given they are exponentially linearly related, find an equation to model the data. (0,1) and (15, 20) a = 1 Predict the percent risk for a person with a score of 20

Exponential Functions from Tables A person’s heart attack risk can be estimated by using Framingham point scores. Men’s risk of having a heart attack in the next 10 years is shown in the table. Framingham Score Risk) (percent 1 5 2 10 6 15 20 17 30

Plot a Scattergram

Chose two points Divide the equations Find a (5,2) (15,20) b represents the percent risk grows exponentially by 26% for each score point a represents the approximate initial percent risk at a score of 0 points

Graph line over scattergram to check