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Aug 15, 2024
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# MATH 159 - College Algebra

4 Credit: (4 lecture, 0 lab, 0 clinical) 4 Contact Hours: [Math Level 5 ]

This course explores the concept of functions as models of change. Functions studied include linear, piecewise defined, quadratic, inverse, exponential, logarithmic, power, polynomial and rational. Other topics included are modeling, concavity, transformations of functions, compositions and combinations of functions, sequences and series. This course stresses conceptual understanding and multiple ways of representing mathematical ideas.
OFFERED: spring and fall semesters

Course Goals/ Objectives/ Competencies:
Goal 1:  Students will investigate and review the concept of a function and extend their understanding of linear functions.

Objectives:  The student will

1. identify functions from different representations of data and understand basic function notation.
2. compute the average rate of change of a function on an interval and explain what this computation represents.
3. describe what quality must exist for given data to represent a linear function and write a linear function to model the data.
4. interpret what the slope and y-intercept of a linear function represent in different contexts.
5. write the equations of horizontal, vertical, parallel, and perpendicular lines with given information.
6. determine where two lines intersect and interpret this point in different contexts.
7. use linear regression to find the line of best fit for a data set, interpret the parts of the equation, and use the equation for data analysis.

Goal 2:  Students will deepen their understanding of functions and become familiar with several types of functions.

Objectives:  The student will

1. evaluate functions and interpret the results using tables, graphs, equations, and verbal explanations.
2. determine the domain and range of a function from equations or graphs.
3. evaluate and graph piecewise defined functions and write equations of piecewise defined functions to model situations.
4. use inverse function notation, interpret this notation in the context of a problem, and be able to find inverses for basic functions.
5. determine the concavity of a function by examining the rate of change of the function.
6. determine the zeros of a quadratic function using several methods.
7. use average rate of change to describe the concavity of a quadratic function.

Goal 3:  Students will become familiar with exponential functions.

Objectives:  The student will

1. identify the form of an exponential function and write the equation for this type of function when the growth or decay rate is known.
2. determine the percentage growth/decay rate from the formula of an exponential function.
3. distinguish between linear and exponential functions by looking at equations, graphs, and tables.
4. write formulas for exponential functions based upon certain data points.
5. classify the different geometric qualities of exponential functions.
6. compute compound interest for different types of compounding including continuous and determine the effective rate based on the type of compounding used.
7. set up and solve application problems involving exponential functions.

Goal 4:  Students will investigate logarithmic functions.

Objectives:  The student will

1. state the definition of logbx.
2. write an exponential expression equivalent to a given logarithm and vice versa.
3. use the definition of logbx to evaluate certain logarithm expressions without a calculator.
4. compute logbx using the common logarithm or natural logarithm.
5. use properties of logarithms to rewrite expressions and solve equations involving logarithms.
6. use logarithms to solve exponential equations.
7. solve different application problems such as half life and doubling time with logarithms.
8. rewrite a function of the form Q = abt in the form Q = aekt and vice versa.
9. state the domain and range for any logarithmic functions based on the definition of logbx.
10. identify geometric qualities of logarithmic functions.
11. describe the relationship between logarithmic functions and exponential functions.
12. plot numbers on a logarithmic scale.
13. find an appropriate exponential function to model a data set by linearizing the data set using logarithms.

Goal 5:  Students will investigate the relationship between changes made to the formula of a function and changes made to its graph.

Objectives:  The student will

1. sketch y = g(x) + k, y = g(x + k) given the graph of y = g(x).
2. write the equation of a graph that is a translation of a given graph.
3. describe how the graphs of y = -f(x) and y = f(-x) are related to the graph of y = f(x).
4. describe what it means for a function to be even or odd and determine this both graphically and algebraically.
5. describe the relationship between f(x) and g(x) if g(x) = kf(x).
6. describe the relationship between f(x) and g(x) if g(x) = f(kx).
7. sketch the graph of a function obtained by a sequence of transformations of a given function and write a formula for the graph.
8. complete the square to transform a quadratic function into vertex form: y = a(x - h)2 + k.
9. use the vertex form of a quadratic function to describe the function in terms of the transformations of the parent function y = x2.
10. write equations for quadratic functions based on given information.
11. find the zeros and maximum/minimums of a quadratic function using a variety of techniques.
12. use the vertex form of a quadratic function to determine if the function will have 0, 1, or 2 real zeros.
13. determine the non-real zeros of a quadratic function.
14. perform arithmetic calculations with complex numbers.

Goal 6:  Students will investigate different operations on functions.

Objectives:  The student will

1. evaluate a composition of functions given tables of values of the functions.
2. find an expression for the composition of functions.
3. express a function as the composition of simpler functions.
4. identify invertible functions.
5. use properties of inverses to decide if two functions are inverses.
6. state the domain and range of a function and its inverse.
7. state the geometric relationship between a function and its inverse.
8. algebraically combine functions using tables, formulas, and graphs.

Goal 7:  Students will study polynomial and rational functions.

Objectives:  The student will

1. translate “is proportioned to” into a mathematical equation.
2. identify the form of a power function.
3. describe geometric qualities of power functions.
4. find equations for power functions based on given data points.
5. identify the form of a polynomial function.
6. determine the long run behavior of polynomial functions.
7. sketch the graph of a polynomial without a calculator if the zeros and degree of the polynomial are known.
8. write a formula for a polynomial given its graph.
9. identify the form of a rational function.
10. determine geometric qualities of rational functions.
11. determine the zeros and asymptotes of rational functions.
12. write a formula for a rational function given its graph.
13. compare the long-term behavior of power, exponential, and logarithmic functions.
14. determine if a power function or exponential function fits a particular data set and find this function.

Goal 8:  Students will explore sequences and series.

Objectives:  The student will

1. describe and analyze sequences.
2. find a term for a sequence given a formula for the nth term.
3. identify arithmetic and geometric sequences.
4. write and apply the formula for the nth term of a arithmetic or geometric sequence.
5. find and interpret the partial sum of arithmetic series or geometric series.
6. write the formula for a sum in sigma notation.
7. find and interpret the infinite sum of a geometric series
8. compare finite and infinite geometric series

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