This course is the study of functions from both an algebraic and graphical approach. The relationship between the equation form of a function and its graph will be emphasized enabling students to develop and analyze mathematical models using these functions. In this course students must begin to make the transition from practicing the mechanics of intermediate algebra to using these skills in the exploration of mathematical ideas.

Functions and Their Graphs

Required Core Outcomes

1.             Describe a function including domain, range, independent and dependent variables. Use functional notation in your discussion.

2.             Understand and correctly use functional notation. Simplify functional expressions including difference quotients.

3.             Know the basic shapes of  without a graphing calculator.

4.             Find zeros of a function and understand the relationship between the zeros of the function and the x-intercepts of its graph, , and the solutions to .

5.             Determine if a function is one-to-one from its graph. Know the symmetry of even and odd functions.

6.             Combine two or more functions by adding, subtracting, multiplying, dividing, and composing. Find domains of each of the resulting functions.

7.             Find inverses of one-to-one functions and verify using composition. Understand the relationship between and .

8.             Without a calculator interpret and describe the graphical modification of a basic graph when is composed with a linear function. Include vertical and horizontal shifting, vertical and horizontal stretching or compressing, and reflecting about an axis or the origin.

Strongly Recommended Outcomes

9.             Determine from the graph of a function whether the function is continuous, and estimate intervals where the graph is increasing or decreasing.

10.           Decompose a function, that is, find two functions whose composition is a given function.

11.           Sketch graphs of piecewise-defined functions including absolute value.

Optional

12.           Sketch graphs of step functions.

13.           Use a graphing utility to verify hand-graphs of all functions introduced in Math 4.              

Polynomial Functions

Required Core Outcomes

1.             Create a polynomial function or model from an equation or formula that involves two or more unknowns by substituting other given or known information. (Examples: Write the function used to maximize the area in a corral given a fixed length of fence. Express the area of a circle as a function of the diameter.)

2.             Factor nth degree polynomials with real coefficients into a product of first and second-degree polynomials. Methods to achieve this goal may include a combination of familiar factor methods, technology, or the Rational Root Theorem. (The use of polynomial long division or synthetic division must be stressed. Do not rely upon “factor functions” in computers or calculators.) The above skills should result in a practical working knowledge of the Fundamental Theorem of Algebra.

3.             Find the zeros of a polynomial function , understand the relationship between the zeros of f, the x-intercepts of the  graph, and the solutions to the equation . Understand the relationship between repeated zeros and the number of distinct x-intercepts of the graph.

4.             Sketch polynomial graphs without the aid of a graphing utility. Use the sign of the leading coefficient, the degree of the polynomial, knowledge of the existence of relative minima and maxima for higher degree polynomials, symmetry tests, behavior at the intercepts, and the graph’s end behavior.

5.             Use multiple approaches to solve polynomial inequalities, including both graphing and sign charting methods.

Rational Functions

Required Core Outcomes

1.             Without the use of a graphing calculator, sketch the graph of a rational function by knowing its domain, locating its intercepts, knowing the affect of multiplicity of x-intercepts, locating its vertical and horizontal asymptotes, determining whether it crosses its horizontal asymptotes, identifying its holes, and plotting additional points as necessary.

2.             Use multiple approaches to solve rational inequalities, including both graphing and sign charting methods.

Strongly Recommended Outcomes

3.             Solve application problems in which a rational model is given.

4.             Create a rational model or function to solve an application problem.

5.             Create a rational model or function that satisfies a given graph or given information about asymptotes, intercepts, domain, etc.

Optional

6.             Locate vertical asymptotes of a rational function whose denominator is a polynomial of degree 3 or more.

7.             Find approximate values of maximum/minimum values of a rational function using graphing technology and use this knowledge to solve applications.

8.             Use a graphing utility to verify hand graphs of rational functions.

Exponential and Logarithmic Functions

Required Core Outcomes

1.             Without the use of a calculator discuss the behavior of basic exponential and logarithmic functions using the following criteria: increasing, decreasing, domain, range, x and y- intercepts without the use of a calculator. These functions include: , , , , , and those formed by the composition of the above functions with linear functions.

2.             Know the relationship between exponential and logarithmic functions as inverse functions (algebraically and graphically), and the composition of these two functions. (ex. )

3.             Be able to change the base of a logarithmic function to base 10 or base in order to approximate the functional values and graph the function using a graphing calculator.

4.             Given a logarithmic or exponential equation (or model), use properties of logarithms, exponents, and knowledge of inverse functions to solve the equation for the indicated variable.

5.             Given information about a growth or decay exponential model, determine the modeling function and use it to predict information.

Strongly Recommended Outcomes

6.             Given a logarithmic or exponential modeling function, use a graphing calculator to graph it, to estimate the intercepts, and interpret their significance.

Optional

7.             Solve logarithmic or exponential inequalities using properties and multiple approaches.

Trigonometry

Required Outcomes

1.             Know the relationship between degree and radian angular measure. Be able to convert from one form to the other.

2.             Develop competency working with reference angles in anticipation of the skills needed to solve trigonometric equations, to use the Law of Sines, and to work with trigonometric functions in other future applications.

3.             Know the definition of the six basic trigonometric functions and find exact functional values of special angles using reference angles, and quadrantal angles. Find approximate functional values using a calculator.

4.             Without the use of a graphing calculator, sketch graphs of the six basic trigonometric function with detailed graphing for sine, cosine, and tangent including horizontal shifts (phase shifts), horizontal stretching or compressing (period), vertical stretching or compressing (amplitude for sine and cosine) as well as vertical shifts.

5.             Develop an understanding and an ability to use the basic trigonometric identities (including the reciprocal, ratio, squared/Pythagorean, sum and difference/expansion, double, and half-angle/power reduction identities). Manipulate trigonometric expressions using algebra and the trigonometric identities.

6.             Understand the development of inverse trigonometric functions including their relationship to the trigonometric functions. Evaluate inverse trigonometric functions exactly as well as with a calculator in radians or degrees without conversion.

7.             Solve trigonometric equations of linear, quadratic, or other algebraic forms using algebraic methods and trigonometric identities. When these equations have exact solutions solve without using a calculator and give approximate answers over specified domains.

8.             Solve triangles and application problems using right triangle trigonometry, the Law of Sines and/or the Law of Cosines.

Strongly Recommended Outcomes

9.             Compare the graphs of the cofunctions.

Advance Algebra Skills

Required Outcomes

1.             Factor expressions with fractional and negative exponents.

2.             Complete the square in various contexts. (example: Write a quadratic function in the form .)

3.             Express vectors in component and trigonometric form and convert between forms.

Strongly Recommended

4.             Write a complex number in polar form.

Optional

5.             Find nth powers of complex numbers using DeMoivre’s Theorem.

6.             Find nth roots of complex numbers.

7.             Use vectors to solve an application problem (example: Given wind magnitude/direction and plane airspeed/direction, find resultant ground speed and direction.)