Open Resources for Community College Algebra

Functions

1. Determine whether a relation is a function when the given relation is expressed algebraically, graphically, numerically and/or within real-world contexts by applying the definition of a function.

2. Domain and Range

   1. Use the definition of domain and range to determine the domain and range of functions represented graphically, numerically, and verbally.

   2. Determine the domain of a function given algebraically.

   3. State the domain and range in both interval and set notation.

   4. Understand how the context of a function used as a model can limit the domain and the range.

3. Function Notation

   1. Evaluate functions with given inputs using function notation where functions are represented graphically, algebraically, numerically and verbally (e.g. evaluate \(f(7)\)).

   2. Distinguish between different expressions such as \(f(x+2)\text{,}\) \(f(x)+2\text{,}\) \(3f(x)\text{,}\) and \(f(3x)\text{,}\) and simplify each.

   3. Interpret \(f(a)=b\) in the appropriate context (e.g. interpret \(f(3)=5\) where \(f\) models a real-world function) and understand that \(f(2)\) is a number not a point.

   4. Solve function equations where functions are represented graphically, algebraically, numerically and verbally (e.g. solve \(f(x)=b\) for \(x\)).

Factoring Polynomials

1. Factor the greatest common factor from a polynomial.

2. Factor a polynomial of four terms using the grouping method.

3. Factor trinomials that have leading coefficients of 1.

4. Factor trinomials that have leading coefficients other than 1.

5. Factor differences of squares.

Rational Functions

1. Determine the domain of rational functions algebraically and graphically.

2. Simplify rational functions, understanding that domain conditions lost during simplification must be noted.

3. Perform operations on rational expressions (multiplication, division, addition, subtraction) and express the final result in simplified form.

4. Simplify complex rational expressions. E.g. \(\frac{\frac{x^2-4}{x^2+x-2}}{x-2}\text{,}\) \(\frac{x^2-9}{\frac{3x+5}{4x+10}}\text{,}\) and \(\frac{\frac{x^2-9}{x^2-x-12}}{\frac{6+2x}{x^2-7x+12}}\text{.}\)

Solving Equations and Inequalities Algebraically

1. Solve quadratic equations using the zero product principle.

2. Solve quadratic equations that have real and complex solutions using the square root method.

3. Solve quadratic equations that have real and complex solutions using the quadratic formula.

4. Solve quadratic equations that have real and complex solutions by completing the square (in simpler cases, where \(a=1\text{,}\) and \(b\) is even).

5. Solve rational equations.

6. Solve absolute value equations.

7. Solve equations (linear, quadratic, rational, radical, absolute value) in a mixed problem set.

   1. Determine how to proceed in the solving process based on equation given.

   2. Determine when extraneous solutions may result. (Consider using technology to demonstrate that extraneous solutions are not really solutions).

8. Check solutions to equations algebraically.

9. Solve a rational equation with multiple variables for a specific variable.

10. Solve applications involving quadratic and rational equations (including distance, rate, and time problems and work rate problems).

    1. Variables used in applications should be well defined.

    2. Conclusions should be stated in sentences with appropriate units.

11. Algebraically solve function equations of the forms:

    1. \(f(x)=b\) where \(f\) is a linear, quadratic, rational, radical, or absolute value function.

    2. \(f(x)=g(x)\) where \(f\) and \(g\) are functions such that the equation does not produce anything more difficult than a quadratic or linear equation once a fraction is cleared or a root is removed if one exists.

12. Solve compound linear inequalities algebraically.

    1. Forms should include:

       1. the union of two linear inequalities (“or” statement).

       2. the intersection of two linear inequalities (“and” statement).

       3. a three-sided inequality like \(a\lt f(x)\lt b\) where \(f(x)\) is a linear expression with \(a\) and \(b\) constants.

    2. Solution sets should be expressed in interval notation.

Graphing Concepts

1. Brief review of graphs of linear functions, including finding the formula of the function given two ordered pairs in function notation.

2. Graph quadratic functions by hand.

   1. Review finding the vertex with the formula \(h=-\frac{b}{2a}\text{.}\)

   2. Complete the square to put a quadratic function in vertex form.

   3. Given a quadratic function in vertex form, observe the vertical shift and horizontal shift from the graph of \(y=x^2\text{.}\)

   4. State the domain and range of a quadratic function.

3. Review finding horizontal and vertical intercepts of linear and quadratic functions by hand, expressing them as ordered pairs in abstract examples and interpreting them using complete sentences in application examples.

4. Solve equations graphically with technology.

5. Explore functions graphically with technology.

   1. Find function values.

   2. Find vertical and horizontal intercepts.

   3. Find the vertex of a parabola.

   4. Create an appropriate viewing window.

6. Graphically solve absolute value and quadratic inequalities (e.g. \(f(x)\lt b\text{,}\) \(f(x)\gt b\)) where \(f\) is an absolute value function when:

   1. given the graph of the function.

   2. using technology to graph the function.

7. Solve function inequalities graphically given \(f(x)\lt b\text{,}\) \(f(x)\gt b\text{,}\) \(f(x)\gt g(x)\text{,}\) and \(a\lt f(x)\lt b\) where \(f\) and \(g\) should include but not be limited to linear functions.