Transformation Interest problems Half life

Slides:

Advertisements

Similar presentations

State the domain and range of each function. 3.1 Graphs of Exponential Functions.

Advertisements

Warm-Up 1) A town with a current population of 701,500 has a growth rate of 1.4 percent. Find the multiplier for the rate of exponential growth. 5 minutes.

Function Families Lesson 1-5.

Exponential and Logarithmic Functions. Exponential Functions Vocabulary – Exponential Function – Logarithmic Function – Base – Inverse Function – Asymptote.

Exponential Functions and their Graphs

Exponential Growth and Decay Functions. What is an exponential function? An exponential function has the form: y = ab x Where a is NOT equal to 0 and.

Exponential Functions Section 1. Exponential Function f(x) = a x, a > 0, a ≠ 1 The base is a constant and the exponent is a variable, unlike a power function.

Graph each function: 1. f(x) = -2x 2 – 4x f(x) = -x 3 + 4x

How does one Graph an Exponential Equation?

Warm Up Simplify. x 1. log 10 x 2. log b b 3w log z 3w3w z 4. b log b (x – 1 ) x – 1.

Logarithms and Exponential Equations Ashley Berens Madison Vaughn Jesse Walker.

Chapter The Natural base, e.

8.2 Day 2 Compound Interest if compounding occurs in different intervals. A = P ( 1 + r/n) nt Examples of Intervals: Annually, Bi-Annually, Quarterly,

Exponential Functions 4.2 Explorations of growth and decay.

Module 4 Section 4.1 Exponential Functions and Models.

1. Given the function f(x) = 3e x :  a. Fill in the following table of values:  b. Sketch the graph of the function.  c. Describe its domain, range,

Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in.

Aim: What is an exponential function?

Exponential Growth Exponential Decay

Exponential and Logarithm

Chapter 8 Exponents and Exponential Functions

Exponential Growth Exponential Decay Graph the exponential function given by Example Graph the exponential function given by Solution x y, or f(x)

Exponential Functions Section 1. Exponential Function f(x) = a x, a > 0, a ≠ 1 The base is a constant and the exponent is a variable, unlike a power function.

SFM Productions Presents: Another saga in your continuing Pre-Calculus experience! 3.1Exponential Functions and their Graphs.

1.6 Shifting, Reflecting and Stretching Graphs How to vertical and horizontal shift To use reflections to graph Sketch a graph.

Exponential Functions L. Waihman A function that can be expressed in the form A function that can be expressed in the form and is positive, is called.

Compound Interest 8.2 Part 2. Compound Interest A = final amount P = principal (initial amount) r = annual interest rate (as a decimal) n = number of.

ACTIVITY 36 Exponential Functions (Section 5.1, pp )

Section 4.1 Exponential Functions

Exponential Functions Section 1.3. Exponential Functions f(x) = a x Example: y 1 = 2 x y 2 = 3 x y 3 = 5 x For what values of x is 2 x

Quiz 7-1: 1. Where does the graph cross the y-axis? 2. f(1) = ? 3. Horizontal asymptote = ? 4. How was the function transformed to get f(x) above? to get.

Graph Horizontal Asymptote: y = -1 Domain: Range:y > -1 all real numbers a = ____ 2 b= ____ 3 h = ____ 2 k = ____.

Journal: Write an exponential growth equation using the natural base with a horizontal asymptote of y=-2.

6.6 The Natural Base, e Objectives: Evaluate natural exponential and

1 Factoring Practice (5 questions). 2 Factoring Practice (Answers)

State the domain and range of each function Exponential Growth and Decay.

Exponential Functions Section 5.1. Evaluate the exponential functions Find F(-1) Find H(-2) Find Find F(0) – H(1)

Exponential Functions and Their Graphs

Exponential Functions and Their Graphs Digital Lesson.

Exponential Functions Evaluate Exponential Functions Graph Exponential Functions Define the number e Solve Exponential Equations.

Exponential Graphs Equations where the variable (x) is the POWER y = ab x – h + k h moves the graph horizontally k moves the graph vertically.

Exponential and Logarithmic Functions

Exponential Functions Algebra III, Sec. 3.1 Objective Recognize, evaluate, and graph exponential functions.

9x – 7i > 3(3x – 7u) 9x – 7i > 9x – 21u – 7i > – 21u

Exponential Functions and Their Graphs/ Compound Interest 2015/16.

GPS: MM3A2e, MM3A2f, MM3A2d.  MM3A2e – Investigate and explain characteristics of exponential and logarithmic functions including domain and range, asymptotes,

SECTION 4.3 EXPONENTIAL FUNCTIONS EXPONENTIAL FUNCTIONS.

Section 3.1 Exponential Functions. Definition An exponential function is in the form where and.

5.2 Exponential Functions and Graphs. Graphing Calculator Exploration Graph in your calculator and sketch in your notebook: a) b) c) d)

The Natural Base, e 4-6 Warm Up Lesson Presentation Lesson Quiz

Find the amount after 7 years if $100 is invested at an interest rate of 13% per year if it is a. compounded annually b. compounded quarterly.

GRAPHING EXPONENTIAL FUNCTIONS f(x) = 2 x 2 > 1 exponential growth 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0.

8.2 Properties of Exponential Functions An education isn't how much you have committed to memory, or even how much you know. It's being able to differentiate.

6.2 Exponential Functions Objective: Classify an exponential function as representing exponential growth or exponential decay. Calculate the growth of.

8.1 Exponential Growth 8.2 Exponential Decay. Exponential Function An exponential function has a positive base other than 1. The general exponential function.

Warm Up  Complete the Grok Activity on the back of your homework (the one with people at the top)

Modeling Constant Rate of Growth (Rate of Decay) What is the difference between and An exponential function in x is a function that can be written in the.

10.2 Exponential and Logarithmic Functions. Exponential Functions These functions model rapid growth or decay: # of users on the Internet 16 million (1995)

Graphs of Exponential Functions. Exponential Function Where base (b), b > 0, b  1, and x is any real number.

Quiz 7-1,2: 1. Where does the graph cross the y-axis? 2. f(1) = ? 3. Horizontal asymptote = ? 4. How was the function : transformed to get f(x) above?

Section 8-2 Properties of Exponential Functions. Asymptote Is a line that a graph approaches as x or y increases in absolute value.

Transforming Linear Functions

How does one Graph an Exponential Equation?

exponential functions

Graphing Exponential Functions

2.7 Graphing Absolute Value Functions

6.9 Graphing Exponential Equations

2.7 Graphing Absolute Value Functions

7.4 Graphing Exponential Equations

Presentation transcript:

Transformation Interest problems Half life 3.1 Exponential Function Transformation Interest problems Half life

Exponential Function f(x) = ax a > 0 and a ≠ 1 x is any real number. Notice 20 = 1 Any number to the zero power (except 0) equals 1

What if there is x is a negative number We would have a decreasing exponential function

What is the horizontal asymptote? We would have a decreasing exponential function

Transformations (shifts happen) y = 4x y = 4x + 3

Now a little to the right y = 4x y = 4(x – 3)

Reflect across the y axis y = 4x y = 4-x

Lets use a different base What do you see?

How about the base e e ≈ 2.71828 ……… it is transcendental. e ≈ Just another base that most natural event follow

Base 10 is also common, that it is on your calculators

Continuous Compounding Interest uses e Now for my favorite equation. Pert P is for principle r is for Rate t is for Time $4000 at 8% interest for 10 years at continuous compounding interest. 4000e(.08)(10) =

Continuous Compounding Interest uses e Now for my favorite equation. Pert P is for principle r is for Rate t is for Time $4000 at 8% interest for 10 years at continuous compounding interest. 4000e(.08)(10) = $8902.16

Compounding a number of times per year equation Amount = N is the number of times per year Quarterly is 4 Semi Annual is 2 Monthly is 12

Decay function In a decay function a (the base) is 0 < a< 1 In other words a fraction The most famous is half life, where a is ½. h is half life P initial amount A is remaining amount

Homework Page 206 – 208 # 2, 7, 13, 21, 33, 43, 47, 51, 54, 57, 61, 67

Homework Page 206 – 208 # 4, 10, 17, 27, 37, 45, 49, 53, 55, 58, 62, 78