Algebra and Functions

Objective

A. Algebra Students should be able to:

1. perform operations of addition, subtraction, multiplication and division of polynomial and rational expressions

2. factorize polynomial expressions, of degree less than or equal to 4, leading to real linear factors

3. apply the Remainder Theorem

4. use the Factor Theorem to find factors and to evaluate unknown coefficients

B. Quadratics Students should be able to:

1. express the quadratic function ax² + bx + c = 0 in the form a(x + h)² + k = 0 , where h and k are constants to be determined

2. determine maximum or minimum values and range of quadratic functions by completion of the square

3. sketch the graph of the quadratic function, including maximum or minimum points

4. determine the nature of the roots of a quadratic equation

5. solve equations in x reducible to a quadratic equation, for example, x⁴ - 6x² + 8 = 0 , etc.

6. use the relationship between the sums and products of the roots and the coefficients of ax² + bx + c = 0

7. solve two simultaneous equations in 2 unknowns in which one equation is linear and the other equation is quadratic

C. Inequalities Students should be able to:

1. find the solutions sets of quadratic inequalities using algebraic and graphical methods

2. find the solution sets of inequalities of the form

Study Resources

D. Functions Students should be able to:

1. use terms related to functions

2. determine the range of a function given its domain

3. determine whether a given function is many-to-one or one-to-one

4. determine the inverse of a given function (if it exists)

5. plot and sketch functions and their inverses (if they exist)

6. state the geometrical relationship between the function y = f(x) and its inverse f ^-1(x)

7. find the composition of two functions

8. recognize that, if g is the inverse of f, then f [g(x)] = x , for all x, in the domain of g

E. Surds, Indices, and Logarithms Students should be able to:

1. perform operations involving surds

2. use the laws of indices to solve exponential equations with one unknown

3. use the fact that log_ab = c implies that a^c = b where a is any positive whole number

4. simplify expressions by using the laws:

• Practice Sheet: Indices and Logs
• Practice Sheet: Indices and Surds