MATH 180 - Precalculus

5 Credit: (5 lecture, 1 lab, 0 clinical) 6 Contact Hours:

Objectives:  

1. recognize when relationships are functions.                          
2. use and interpret function notation.
3. represent and analyze functions.
4. evaluate functions and interpret the results.
5. determine the domain and range of a function and recognize restrictions in applied problems.
6. recognize if a function is increasing or decreasing.
7. calculate and interpret the average rate of change.
8. determine if a function is increasing or decreasing using the average rate of change.
9. determine if a function is linear using the average rate of change.
10. describe the concavity of a function using the average rate of change.

Goal 2:  Apply and analyze piecewise functions, inverse functions, composition of functions and transformations of functions.

Objectives:  

1. evaluate, graph, write the formula for and determine the domain and range of piecewise, inverse, composition of and transformations of functions.
2. model real-world applications with piecewise, inverse, composition of and transformations of functions.
3. decompose functions.
4. use inverse function notation and interpret this notation in the context of a problem.
5. distinguish between invertible and noninvertible functions.
6. verify two functions are inverses algebraically through composition.
7. evaluate combinations of functions (determine the sum, difference, product and quotient of two functions defined by formulas, graphs and tables).
8. describe, graph and write a formula of a function given analytically, graphically, numerically or verbally that has multiple transformations.

Goal 3:  Investigate quadratic, power, polynomial and rational functions.

Objectives:  

1. derive formulas for quadratic functions in standard, factored and vertex form, depending on information given, and convert between these forms.
2. model using direct and inverse variation.
3. write formulas for quadratic, power, polynomial and rational functions and describe their unique graphical characteristics.
4. analyze characteristics of quadratic, power, polynomial and rational functions.
5. model and solve application problems involving quadratic, power, polynomial and rational functions.
6. find the inverses of invertible polynomial and rational functions and when necessary, restrict the domain to find the inverse.
7. analyze characteristic of radical functions.
8. analyze limit notation and find limits using a graph or a table.
9. use limits to describe the asymptotic behavior of rational functions.

Goal 4:  Investigate exponential functions.

Objectives:  

1. write, graph, solve and analyze equations for exponential functions given graphically, numerically, or verbally.
2. model and solve application problems involving exponential functions.
3. model compound interest with exponential functions.
4. identify and analyze the nominal and effective annual rates for compounded interest.
5. write and analyze a formula for continuous exponential growth and decay.

Goal 5:  Investigate logarithmic functions.

Objectives:  

1. define log_bx.
2. convert between logarithmic and exponential equations.
3. apply properties of logarithms.
4. use logarithms to solve exponential equations.
5. solve logarithmic equations.
6. convert between functions of the form Q = ab^t in the form Q = ae^kt and identify what each of the parameters represent.
7. graph logarithmic functions and identify qualities of the functions including domain and range.
8. demonstrate an understanding of logarithmic scales.
9. model data with exponential, logarithmic, and logistic functions using regression.  
10. use one-sided limits to describe asymptotic behavior of logarithmic functions.

Goal 6:  Investigate sequences and series.   

Objectives:  

1. identify sequences as arithmetic, geometric or neither.
2. describe, analyze and write sequences.
3. find and interpret the partial sum of arithmetic series or geometric series using the appropriate formula.
4. compare finite and infinite series.
5. write the formula for a series, both finite and infinite, using sigma notation.
6. find the sum of an infinite geometric series using the appropriate formula.
7. use finite and infinite geometric series to model real-world situations and interpret the sums in context.

Goal 7:  Investigate trigonometric functions.

Objectives:

1. define and sketch an angle in standard position and determine coterminal angles.
2. define an angle that has a measure of 1 radian.
3. convert between radians and degrees.
4. calculate the length of an arc.
5. calculate linear and angular speed.
6. write the unit circle definition of the six trigonometric functions.
7. determine the exact values of the six trigonometric functions for common acute angles and for non-acute common angles using reference angles.
8. recognize and use fundamental identities.
9. identify the domain and range of the six trigonometric functions.
10. apply the trigonometric functions to right triangles.

Goal 8:  Explore graphs of trigonometric functions.

Objectives:

1. graph and analyze the basic sine and cosine functions and transformations of these functions.
2. model and analyze formulas for sinusoidal functions that are given graphically, numerically, and verbally including their use in applications such as modeling periodic behavior and harmonic motion.
3. graph and analyze the other 4 trigonometric functions.
4. examine and use the inverse sine, cosine, and tangent functions.
5. find the exact value of expressions involving the inverse sine, cosine, and tangent functions.

Goal 9:  Explore trigonometric identities, equations and other applications.

Objectives:

1. verify the fundamental identities and simplify expressions using the identities.
2. apply the double angle identities to rewrite and simplify expressions.
3. apply the sum and difference identities to rewrite and simplify expressions.
4. solve linear, quadratic, and multiple angle trigonometric equations using algebra and identities.
5. solve trigonometric equations using technology.
6. apply the law of sines and the law of cosines to solve oblique triangles and to model and solve applied problems.
7. identify, describe and analyze the ambiguous case.
8. find the area of an oblique triangle.

Goal 10:  Investigate parametric equations.

Objectives:

1. parameterize curves given analytically and graphically.
2. eliminate the parameter from parametric equations.
3. covert between parametric and rectangular equations.
4. graph and describe the motion given by parametric equations.

*Additional topics to be covered as time allows:  Complex numbers and polar coordinates.