A18-6 exponential fxns notes
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An exponential growth function is one where the rate of change increases by the same factor over equal time intervals, such as yearly interest on a bank account increasing at 8% each year or a rabbit population tripling annually. Exponential growth can be modeled by the function y=C(1+r)t, where C is the initial amount, r is the growth rate, and t is time. To graph an exponential function, make a table of x-y pairs of values, plot the points on a graph, and draw a curve through the points that best models the relationship.
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![Algebra 1<br />8.5Exponential Growth Functions<br />Get ready:What do you think an exponential growth function is?<br />Get set:Evaluate when x = 5.<br />a)-12+9x b)-12+9x<br />What is the difference?<br />Go!<br />Exponential Growth<br />Something is growing __________________________ if it increases by the same ______________________ for each unit of time (i.e., year, month, second, etc.).<br />y=C(1+r)t<br />Example 1:You deposit $500 in an account that pays 8% annual interest compounded yearly.<br />--Compounded means that it adds 8% of what is in the account at the end of every year…so it adds more and more every year. That’s how exponential functions work.<br />What is the account balance after 6 years?<br />Example 2:A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.<br />,[object Object]](https://image.slidesharecdn.com/a18-6exponentialfxns-110228160231-phpapp01/85/A18-6-exponential-fxns-notes-1-320.jpg)
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Solution
22)
a)the investment will be =5000*(1+.0125)^20=$6410.186
b)10000<5000(1+.0125)^n
2<(1.0125)^n
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n> 55quaters or n=14 years
c)new investment=5000*(1+5/1200)^60=$6416.793
gain=$6.607
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2. Parallel lines have the same slope and never intersect, so they have no solution. Coinciding lines are the same single line, so they have infinitely many solutions.
3. The document provides examples of solving systems of equations and inequalities through graphing, substitution, and elimination. The solutions are given as points or descriptions of regions.
PC Test 2 study guide 2011
1. The document is a study guide for a precalculus test that covers topics involving vectors, complex numbers, and their representations and operations.
2. It lists 15 problems involving finding magnitudes and direction of vectors, vector components, dot and cross products, complex numbers in rectangular and polar form, and converting between polar and rectangular coordinates.
3. The problems cover basic vector and complex number calculations, representations, properties and conversions.
A2 Test 2 study guide with answers (revised)
The document is an algebra 2 study guide that provides practice problems for graphing and solving various types of inequalities and systems of equations, including:
1) Graphing linear and compound inequalities on number lines and coordinate planes
2) Solving absolute value equations and inequalities
3) Solving systems of linear equations through graphing, substitution, and elimination
The study guide contains over 50 problems to help students prepare for an algebra 2 test.
A2 Test 2 study guide with answers
This document provides a study guide with answers for an Algebra 2 Test 2. It includes instructions to graph and solve various types of inequalities, systems of equations and absolute value equations, including linear and compound inequalities, systems of linear equations solved by graphing, substitution and elimination, and absolute value equations and inequalities.
A1 Test 2 study guide with answers 2011
The document is an algebra 1 test study guide containing 30 problems involving graphing and solving various types of inequalities, including linear, compound, absolute value, and graphing solutions on number lines. Key topics covered include writing inequalities from graphs, solving single-variable and compound inequalities, graphing linear inequalities on a coordinate plane, and solving absolute value equations and inequalities.
A1 Test 2 study guide
1. The document provides a study guide for an Algebra 1 Test 2 with questions about: graphing and writing inequalities on number lines, solving linear inequalities and compound inequalities, graphing linear inequalities on the xy-plane, and solving absolute value equations and inequalities.
2. Questions involve skills like graphing inequalities, writing inequalities from graphs, solving one-step and two-step inequalities, and solving absolute value equations and inequalities.
3. The study guide covers key Algebra 1 concepts to help prepare for Test 2.
PCExam 1 practice with answers
1. This practice exam covers topics like complex numbers, functions, limits, and graphing.
2. It asks students to choose problems involving adding, multiplying, composing, and finding inverses of various functions like f(x)=9x^2+1 and g(x)=x-1.
3. Students also must graph and classify functions, evaluate limits, and perform operations on complex numbers, plotting them on a plane. The exam covers concepts in precalculus.
PCExam 1 study guide answers
1. The document provides examples and definitions for various mathematical concepts including natural numbers, integers, rational numbers, real numbers, imaginary numbers, and complex numbers.
2. It also includes problems involving composition of functions, finding inverses of functions, operations on complex numbers, graphing and classifying functions, and evaluating limits.
3. Examples cover topics like composition and inverses of functions, operations on complex numbers, classifying functions as linear, quadratic, constant, etc. and their domains and ranges, and evaluating limits including ones that are undefined.
PC Exam 1 study guide
This study guide covers topics in precalculus including:
1) Examples of natural numbers, integers, rational numbers, and real numbers.
2) Composition of functions including finding f(g(x)) and g(f(x)) for given functions f(x) and g(x).
3) Finding inverses of functions and verifying inverses by composition.
4) Operations on complex numbers including plotting, finding modulus, distance, midpoint, addition, subtraction, multiplication, and division.
5) Graphing functions, classifying function types, and stating domains and ranges.
6) Finding limits of functions as x approaches values.
PC 1 continuity notes
A continuous function is defined as one that is defined at a point c, where the limit exists and the function approaches the same y-value from both sides of point c. A function may be discontinuous due to an infinite jump, a jump, or a point discontinuity. Rational functions are defined as the quotient of two polynomial functions. They have asymptotes which occur where the denominator is zero, creating either a vertical or horizontal asymptote in the graph. The end behavior of functions can be determined by considering whether they are increasing or decreasing over different intervals.
PC 1 continuity
This document discusses continuity and discontinuities in functions, defining continuous functions as smooth curves where the domain includes all real numbers. It presents tests for continuity, requiring a function to be defined at a point, have an existing function value, and approach the same y-value from both sides. Critical points and extrema are defined as maximums where a function changes from increasing to decreasing, and minimums where it changes from decreasing to increasing. Rational functions are described as the quotient of two polynomial functions, with limited domains and possible vertical asymptotes.
A1 3 linear fxns
The document discusses the slope-intercept form of a linear equation, y=mx+b. It explains that x represents the input, y represents the output, m is the slope of the line, and b is the y-intercept. It provides instructions for graphing a linear equation in slope-intercept form, which is to first plot the y-intercept and then use the slope to rise and run to the next point, connecting the points with a line.
A1 3 linear fxns notes
The document discusses slope and linear equations. It defines slope as rise over run and provides examples of calculating slope from graphs and ordered pairs. It also defines the standard form of a linear equation as y=mx+b, where m is the slope and b is the y-intercept. The document explains how to graph lines from their equations in standard form by plotting the y-intercept and using the slope to rise and run to the next point.
A1 2 linear fxns
This document discusses the slope-intercept form of a linear equation, y=mx+b. It explains that x represents the input, y represents the output, m is the slope of the line, and b is the y-intercept. It provides instructions for graphing a linear equation in slope-intercept form, which is to first plot the y-intercept, then use the slope to rise and run to the next point and connect them with a line.
A1 2 linear fxns notes
This document discusses intercepts and slope of lines in algebra 1. It defines x-intercepts as where the line crosses the x-axis and y-intercepts as where it crosses the y-axis. It provides the equations for finding each, and includes two examples of finding intercepts and graphing lines. It also defines slope as rise over run and asks what the slope is of a given graph, with homework assigned on slope of lines and graphing lines intuitively.
A2 2 linear fxns notes
This document provides instructions and examples for finding the slope and y-intercept of a line from its equation, ordered pairs, or graph and using them to graph the line. It explains that slope is defined as rise over run and is used along with the y-intercept to graph lines by standard form, plotting the y-intercept and using slope to find successive points. Exercises are provided to have students practice finding slope, y-intercept, and graphing lines from different representations.
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A18-6 exponential fxns notes
- 2. What is the population after 5 years?Graphing Exponential Functions<br />Make a ___________________ of values.<br />_______________ the points on graph.<br />Draw a ________________________________________ through the points.<br />Example 3:Graph the exponential growth model from Example 2.<br />time012345Population20<br /> <br />HW p. 480 (10 – 20 even)<br />