21. Give a brief background of the use of logarithm. Tell and explain to the .... Star 55 Cancri has three other planets, making it the first known four planet.
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Unit 2 Unit Title:
Polynomials, Exponential and Logarithmic Function
Subject / Quarter: Key Words:
Mathematics IV / Second Quarter
Year Level:
4th Year
exponential functions, laws of logarithm, graphs of function, zeros, factors, logarithm, growth and decay
Designed by:
Ms. Maria Heizel T. Salayo
School District:
Quezon City District 2
Time Frame: School:
TBD
New Era University
Unit Summary In this unit students will learn about the Polynomials, Exponential and Logarithmic Function. Students are introduced to nonlinear functions. Students first learn about polynomials and operations involving monomials and polynomials. In order to delve into the physical and life sciences we need to compactly describe and compare the extremes in deep time and deep space. In this unit, we introduce the tools that scientist use to present very large and very small quantities. Exponential and logarithmic functions are used to describe quantities that change over time. Exponential functions can model such diverse phenomena as bacteria growth, radioactive decay, compound interest rates, inflation, musical pitch, and family trees. If we know a specific output for an exponential function, we can find the associated input by the use of logarithmic functions, the close relatives of exponential functions.
Big Idea for this Unit Many situations in the real world cannot be modeled with first-degree equations. They require the use of functions that are not linear. Students will learn about exponential equations. They will learn the laws of logarithms and how to apply them to help simplify expressions and solve equations. Use polynomial, exponential and logarithmic functions to model real world phenomena and solve problems arising from those situations. Write, solve and/or graph linear, polynomial, exponential and logarithmic equations and/or inequalities to model relationships between quantities.
Unit Design Status:
Completed template pages – Stages 1, 2, and 3
Completed rubrics
Completed blueprint for each performance task
Suggested extensions
Directions to students and teachers
Suggested projects
Suggested accommodations
Differentiated instructions
Status: Initial draft (date 01/13/10) Peer reviewed Content reviewed
Revised draft (date 05/18/10) Field tested Validated Anchored
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STAGE 1: Desired Results Established Goals Content Standards The learner demonstrates understanding of the key concepts and principles of polynomial, exponential and logarithmic functions and their applications to real life. Performance Standards The learner: uses precise mathematical language to explain the characteristics of polynomial functions. performs accurately operations on polynomial functions. applies appropriately polynomial functions to fields such as business, economics, science etc. creates a variety of models to represent situations related to exponential and logarithmic functions. communicates clearly ideas related to exponential and logarithmic functions. investigates situations involving growth and decay.
Essential Understandings The students understand that…
The concepts of polynomial functions can be applied in real life (e.g. business, economics, science) to maximize profits, minimize costs and as basis for decision-making. The concepts of exponential and logarithmic functions can be applied to predict trends in fields such as business, economics, research, science, etc. Algebra represents mathematical situations and structures for analysis and problem solving.
Essential Questions
How are the concepts of polynomial functions applied in the environment?
How are the concepts of polynomial functions used in fields such as business, economics, science, etc.?
How are the concepts of exponential and logarithmic functions applied in real life?
Why do we study exponential and logarithmic functions?
When do quantities have a nonlinear relationship?
Exponential functions are used to model trends in population.
How can functions be used to model real-life situations?
Functions describe the relationship between two variables.
How do we apply exponential and logarithmic functions to predict trends in population?
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STAGE 1: Desired Results (continued)
Graphs of functions that are inverses of each other are reflections across the line y = x.
Quantities such as population and compound interest grow exponentially.
Functions arise from practical situations.
The use graphing calculator as a strategy for investigating the shape and behavior of polynomial functions.
The Fundamental Theorem of Algebra states that, including complex and repeated solutions, an nth degree polynomial equation has exactly n roots (solutions). The following statements are equivalent: - k is a zero of the polynomial function f; - (x – k) is a factor of f(x); - k is a solution of the polynomial equation f(x) = 0; and - k is an x-intercept for the graph of the polynomial.
An exponential equation is an equation with a variable in the exponent.
Properties of logarithms help us identify expressions and solve exponential equations.
Simplifying an expression is different than solving an equation.
Exponential and logarithmic functions are either strictly increasing or strictly decreasing.
Now that I have found my data, how do I keep track of all these numbers?
These things look like just a bunch of numbers, but are there some ways they are related?
I understand these results, but how do I record this data so others can understand the results?
I have all of my results recorded and something has changed, how does this affect my original results?
How can you determine whether a set of data displays exponential behavior?
How can you differentiate between exponential growth and decay?
How do properties of logarithms help us simplify expressions and solve exponential equations?
Why do we need logarithms?
How do logarithms make calculations easier?
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STAGE 1: Desired Results (continued)
Exponential and logarithmic functions can be used to model real-life situations.
Exponential functions and graphs display varying rates of change.
Exponential and logarithmic functions are closely related.
Knowledge The students will know… A. Polynomial Functions 1. identifying a polynomial function from a given set of relations 2. determining the degree of a given polynomial function 3. finding the quotient of polynomials by: algorithm synthetic division 4. finding the value of p(x) for x = k by: synthetic division Remainder Theorem 5. finding the zeros of polynomial functions of degree greater than 2 by: Factor Theorem factoring synthetic division depressed equations 6. graphing the polynomial functions of degree greater than 2 (use graphing calculator if available) B. Exponential and Logarithmic Functions 1. identifying certain relationships in real life which are exponential (e.g. population growth over time, growth of bacteria over time, etc.) graphing of an exponential function f(x) = ax
Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to:
distinguish between relations and functions that are expressed algebraically and graphically.
find the value of a function for a given element from the domain.
identify the zeros of the function algebraically and confirm them, using the graphing calculator.
evaluate polynomial functions using synthetic division
find the quotient and remainder using synthetic division
utilize the remainder and factor theorem
find the real zeros of a polynomial function
graph polynomial functions
determine the linear factors of a polynomial expression when the zeros of the corresponding polynomial function are displayed on a graph.
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STAGE 1: Desired Results (continued) 2. describing some properties of the exponential function, f(x) = ax from its graph a>1 o
