Overview
| Site: | Clare |
| Course: | Michigan Algebra I Sept. 2012 |
| Book: | Overview |
| Printed by: | Guest user |
| Date: | Saturday, November 23, 2024, 11:30 PM |
Description
Introduction
This unit addresses exponential and power functions. Throughout the course, students will be comparing and contrasting power functions; including end behavior, maximum and minimums, and symmetry. They will also compare and contrast characteristics of exponential functions in order to enrich the understanding of exponential growth and decay. Real world situations will be explored, such as, compound interest, carbon-14 dating and bacterial growth. This unit should take approximately 8 to 9 weeks in a traditional hourly schedule.
Students Will Be Able To
After successful completion of this unit, students will be able to understand the concept of exponential and power functions. They will be able to:
- Compare and contrast direct and inverse power functions.
- Compare and contrast end behavior and asymptotes of power functions and exponential functions.
- Compare and contrast maximum and minimum values of power functions.
- Find symmetry of power functions.
- Find inverses and limits to power functions.
- Calculate with rules of exponents.
- Write and evaluate exponential functions that include exponential growth or decay, half-life, and compound interest.
- Look at a situation and determine if it can be modeled by a polynomial function, an exponential function, a power function, or none of the above.
Prior Knowledge
The following GLCEs will be needed to understand this unit:
A.PA.07.09 Recognize inversely proportional relationships in contextual situations; know that quantities are inversely proportional if their product is constant, and that an inversely proportional relationship is of the form y = k/x where k is some non-zero number.
A.RP.07.10 Know that the graph of y = k/x is not a line, know its shape, and know that it crosses neither the x nor the y-axis.
A.RP.08.01 Identify and represent linear functions, quadratic functions, and other simple functions including inversely proportional relationships; cubics; roots; and exponentials; using tables, graphs, and equations.
N.ME.08.02 Understand meanings for zero and negative integer exponents.
Mastered HSCEs
The following Michigan High School Content Expectations will be mastered in this unit.
L1.1.4 Describe the reasons for the different effects of multiplication by, or exponentiation of, a positive number by a number less than 0, a number between 0 and 1, and a number greater than 1.
L2.1.2 Calculate fluently with numerical expressions involving exponents. Use the rules of exponents, and evaluate numerical expressions involving rational and negative exponents, and transition easily between roots and exponents.
A1.1.2 Know the properties of exponents and roots and apply them in algebraic expressions.
A1.2.6 Solve power equations and equations including radical expressions, justify steps in the solution, and explain how extraneous solutions may arise.
A3.2.1 Write the symbolic form and sketch the graph of an exponential function given appropriate information.
A3.2.4 Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and how the base affects the rate of growth or decay.
A3.2.5 Relate exponential functions to real phenomena, including half-life and doubling time.
A3.4.1 Write the symbolic form and sketch the graph of power functions.
A3.4.2 Express directly and inversely proportional relationships as functions and recognize their characteristics.
A3.4.3 Analyze the graphs of power functions, noting reflectional or rotational symmetry.
Addressed HSCEs
The following Michigan High School Content Expectations will be addressed within this unit.
A1.1.1 Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables.
A1.2.1 Write and solve equations and inequalities with one or two variables to represent mathematical or applied situations.
A1.2.8 Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable. Justify steps in the solution.
A2.1.1 Recognize whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function and identify its domain and range.
A2.1.2 Read, interpret, and use function notation and evaluate a function at a value in its domain.
A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations.
A2.1.6 Identify the zeros of a function and the intervals where the values of a function are positive or negative. Describe the behavior of a function as x approaches positive or negative infinity, given the symbolic and graphical representations
A2.1.7 Identify and interpret the key features of a function from its graph or its formula(e), (e.g., slope, intercept(s), asymptote(s), maximum and minimum value(s), symmetry, and average rate of change over an interval).
A2.2.2 Apply given transformations (e.g., vertical or horizontal shifts, stretching or shrinking, or reflections about the x- and y-axes) to basic functions and represent symbolically
A2.2.3 Recognize whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs.
A2.3.2 Describe the tabular pattern associated with functions having a constant rate of change (linear) or variable rates of change.
A2.4.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers.
A2.4.3 Using the adapted general symbolic form; draw reasonable conclusions about the situation being modeled
Source
Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project