MATH165

Functions and Their Graphs

1. Determine if an equation, graph, or table of values represents a function, and state the domain and range of the function.

2. Determine graphically and algebraically whether a function is even, odd, or neither.

3. Test algebraically symmetry with respect to x-axis, y-axis, and the origin.

4. Determine the local minima and local maxima of a function and the interval(s) on which the function is increasing, decreasing and/or constant, given the graph of the function and using a graphing utility.

5. Graph identity, constant, square, cubic, square root, cube root, absolute value, greatest integer, and reciprocal functions of the form 1/x and 1/(x squared), and check using the graphing utility.

6. Analyze and graph functions in terms of translations, reflections, and expansions or contractions.

7. Determine the graph of a function obtained from a series of transformations.

8. Graph piecewise-defined functions and determine their domains and ranges.

9. Form the sum, difference, product, quotient, and composition of two functions and determine their domains, given their equations or graphs.

10. Find and simplify the difference quotient of a function.

11. Relate the difference quotient and average rate of change of a function.

12. Solve absolute value equations and inequalities algebraically and graphically.

13. With the aid of a graphing utility, determine points of maximization or minimization in situations dealing with distances, areas, and volumes of various geometric figures, as well as in real-world business situations involving revenues, costs, and profits.

Polynomial Functions and their Graphs

1. Determine algebraically the intercepts of a polynomial function, and verify graphically.

2. Determine a polynomial function’s intercepts, extreme values, turning points, and intervals of increase and decrease by using a graphing utility.

3. Determine the end behavior of a polynomial function using the degree of the polynomial and its leading coefficient.

4. Construct a polynomial function with specified zeros.

5. Test factors of polynomial functions by using the Remainder and Factor Theorems, and factor polynomial functions to find the real zeros.

6. Solve polynomial equations by using the Rational Zeros Theorem and synthetic division in conjunction with a graphing utility.

7. Find the complex solutions of polynomial equations.

8. Solve polynomial inequalities algebraically and graphically.

Rational Functions

1. Find the domain of a rational function.

2. Find vertical and horizontal asymptotes and oblique asymptotes of the graph of a rational function algebraically, graphically, and numerically.

3. Find the x-intercept(s), y-intercept and coordinates of any hole(s) in the graph of a rational function.

4. Graph a rational function by hand and check it by using a graphing utility.

5. Solve rational equations and inequalities algebraically and graphically.

6. Solve application problems involving rational functions.

Exponential and Logarithmic Functions

1. Determine if a function is one-to-one and find its inverse if possible.

2. Define and graph basic exponential and logarithmic functions (with a strong emphasis on functions with base 10 and e) and their transformations, and identify their domains, ranges, intercepts and asymptotes both by hand and using a graphing utility.

3. Solve exponential and logarithmic equations both algebraically and graphically using a graphing utility.

4. Use exponential and /or logarithmic functions to solve problems involving compound interest and growth and decay.

5. Expand or condense logarithmic expressions using the properties of logarithms.

6. Use the change of base formula to find the logarithm of any base other than 10 or e.

Angles and Trigonometric Functions

1. Sketch a given angle in standard position.

2. Apply the relationship between arc length, area of a sector, central angle, and the radius of a circle and that between linear speed, angular speed, and the radius of a circle.

3. State the unit circle definitions and the domain and range of each of the six trigonometric functions for any real number t.

4. Find exact values of the six trigonometric functions of special real numbers in standard position if given partial information.

5. Evaluate the six trigonometric function of an acute angle of a right triangle.

6. Use the Reciprocal Identities, Quotient Identities, Even-Odd Identities, and the Pythagorean Identities to simplify trigonometric expressions.

7. Sketch the graphs of the of sine, cosine, and tangent trigonometric functions and their transformations both by hand and by using a graphing utility, stating their periods, amplitudes, and phase shifts if applicable.

8. Find an equation for a sinusoidal graph.

Analytic Trigonometry

1. Define the inverse sine, inverse cosine, and inverse tangent functions, and determine their domains and ranges.

2. Evaluate the inverse sine, inverse cosine, and inverse tangent function values exactly when possible and, if not possible, approximate using a graphing utility.

3. Use properties of inverse functions to find the exact value of certain composite functions and solve equations.

4. Determine graphically whether an equation appears to be an identity if so verify algebraically.

5. Use identities, including, Quotient Identities, Reciprocal Identities, the Pythagorean Identities, the Even-Odd Identities, the Sum and Difference Identities, the Double Angle Identities, and the Half Angle Identities for sine, cosine and tangent to simplify expressions.

6. Determine exact solutions of trigonometric equations.

7. Estimate solutions to trigonometric equations by using a graphing utility.