MATH115

Solution of Equations and Inequalities

1. Solve linear equations and linear inequalities in one variable algebraically and graphically.

2. Solve quadratic equations and inequalities algebraically and graphically.

3. Solve rational equations and inequalities algebraically and graphically.

4. Solve radical equations algebraically and graphically.

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

6. Solve polynomial equations and inequalities algebraically and graphically.

Characteristics of 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. Determine graphically if a function is increasing, decreasing, or constant.

4. Determine the location of maxima and minima using a graphing utility.

Basic functions and their transformations.

1. Graph identity, constant, square, cubic, square root, cube root, absolute value, reciprocal functions of the form 1/x and 1/(x squared), and piecewise-defined functions by hand, and check using the graphing utility.

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

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

Operations on Functions

1. Find the sum, difference, product, quotient, and composition of two functions and their domains given their equations or graphs.

2. Evaluate composite functions.

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

Linear Functions and their Graphs

1. Graph linear functions by hand, and check using a graphing utility.

2. Find the intercepts of a linear function algebraically and graphically.

3. Determine the slope of a line joining two points in the plane, and determine the slope of a line given its equation.

4. Produce a graph of a linear function by using slope and a point on a line.

5. Write equations of lines satisfying specified sets of conditions.

6. Write equations of parallel and perpendicular lines satisfying specified sets of conditions.

7. Estimate the equation of the line of best fit for a set of bivariate data algebraically and by using a regression program in a graphing utility.

8. Solve applied problems involving the line of best fit for a set of bivariate data.

Quadratic Functions and their Graphs

1. Graph quadratic functions by hand and check using a graphing utility.

2. Find the intercepts of quadratic functions algebraically and graphically.

3. Identify the vertex and axis of symmetry of a quadratic function both algebraically and graphically (sketch by hand and by using a graphing utility).

4. Convert the standard form of a quadratic function into vertex form.

5. Determine the maximum or minimum value of a quadratic function both algebraically and by using a graphing utility.

6. Solve applied maximum and minimum problems both algebraically and by using a graphing utility.

7. Determine a quadratic model for a given set of data using a graphing utility.

Quadratic Equations and Inequalities

1. Solve quadratic equations algebraically using factoring, the square root property, completing the square, and the quadratic formula, and verify solutions graphically.

2. Perform basic arithmetic operations on complex numbers, and solve quadratic equations with complex solutions.

3. Determine the number and type of solutions to a quadratic equation using the discriminant.

4. Solve equations that are quadratic in form algebraically and verify solutions using a graphing utility.

5. Solve inequalities involving quadratic functions algebraically, and verify solutions using a graphing utility.

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.

Polynomial Equations and Inequalities

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

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

3. Find the complex solutions of polynomial equations.

4. Solve polynomial inequalities algebraically and graphically.

Analytic Geometry and Conic Sections

1. Find the distance between two points in the coordinate plane by means of the distance formula, and find the midpoint of the segment joining the points.

2. Complete the square to write equations of circles and parabolas, vertical and horizontal, in standard form.

3. Determine the center and radius from the equation of the circle, and use these to sketch its graph by hand.

4. Write the equation of a circle satisfying a set of specified conditions.

5. Determine the vertex from the equation of a parabola, and use transformations to sketch its graph.

6. Write the equation of a parabola given its vertex and another point.

Systems of Equations and Inequalities

1. Solve 2 x 2 and 3 x 3 systems of linear equations by the methods of substitution and elimination.

2. Classify systems of equations as inconsistent, consistent, or dependent, and represent solutions of dependent systems.

3. Solve applied problems by using a system of linear equations.

4. Solve linear and nonlinear two-variable systems of inequalities graphically.

5. Solve systems of nonlinear equations graphically and algebraically.

Sequences and Series

1. Write terms of a sequence given its general term.

2. Write the terms of a sequence using a recursive definition.

3. Determine if a sequence is arithmetic, geometric, or neither. (Optional)

4. Given an arithmetic or geometric sequence of terms, write a general formula for the nth term of the sequence. (Optional)

5. Expand a sum of terms represented by summation notation, and express a finite sum in terms summation notation. (Optional)

6. Find the sum of a finite number of terms of a sequence using properties of sequences, and, in particular, find finite sums of arithmetic and geometric sequences. (Optional)

7. Find the sum of a geometric series if it exists. (Optional)

8. Expand powers of binomials.