Unit 7

Units 1 to 6 Review

Read

Part 7.1A corresponds to sections 2.1 to 2.3, page 62; sections 3.1 to 3.4, page 106; sections 9.2 and 9.3, page 446; and sections 7.1 and 7.3, page 334 of your Pre-Calculus 12 textbook.

Radical Function Reminder

Recall radical functions contain a variable under the radical sign. A radical can be written as a power with a rational exponent.

If the radical is an even root, such as a square root or fourth root, the value under the root sign must be greater than or equal to zero.

Also, remember a linear function is related to the square root of that linear function. Strategies were learned to recognize characteristics of the radical function by examining the linear function.

Polynominal Function Reminder

Recall a polynominal function can have many terms that consist of both coefficients and variables of varying whole number degrees. The terms are usually written in descending order by degree. Review the shapes of the various polynomial functions on page 113 of your Pre-Calculus 12 textbook.

Factoring can be used to determine the zeros of a polynomial function. Factoring polynomials of degree three or more may require long division or synthetic division.

The integral zero theorem can be used to make a list of integral values that could be zeros of a polynomial function.

The remainder theorem and factor theorem can then be used to test these integral values to find a binomial factor of the polynomial.

Sketching the graphs of polynomial functions can be done quite accurately by considering the sign of the leading coefficient, the degree, end behaviour, factors, zeros, and multiplicity of the zeros of the function.

Rational Function Reminder

Recall a rational function is a fraction with a variable in the denominator.

Dividing by zero is not defined, so a denominator cannot be equal to zero. The variable in a rational function needs to be restricted so the denominator does not equal zero.

Sometimes rational functions can be simplified by factoring and eliminating factors that appear in both the numerator and denominator. A factor that occurs only in the numerator corresponds to a zero, a factor that occurs only in the denominator corresponds to a vertical asymptote, and a factor that occurs in both numerator and denominator corresponds to a point of discontinuity.

The domain of a rational function can often be determined by looking at non-permissible values. The range can be predicted from the graph of the function, but non-permissible values and horizontal asymptotes may be required to determine the complete range.

Exponential Function Reminder

Recall an exponential function contains a variable in the exponent. When the base of an exponential function is greater than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», it is an exponential growth function. When the base of an exponential function is between «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», it is an exponential decay function.

Examples of situations modelled by exponential functions are financial investments involving interest, population growth, asset depreciation, and population decrease.