Preliminaries and Polynomials¶

Goals for Section 1.1 of the book¶

Students should be able to do each of the following:

Students should be able to do each of the following:

Determine if a given description of a relationship is a function using written, graphical, and tabular descriptions.
Define an algebraic representation of a function given a written description of graphical representation.
Use correct notation to describe a function and be able to identify the order of operations in a given expression.
Determine the x and y intercepts of a function.
Determine the domain and range of a given function.
Compute values of a given function.
Make a rough sketch of the graph of a function.
Determine domain or range given representation of a function.
Determine a function given a description of a situation.
Determine intervals in the domain where a function is increasing/decreasing/constant

Students should be able to do each of the following:

Define a linear relationship from a written description.
Determine the slope of a line from a written description or an algebraic expression.
Compare the slopes of lines and interpret relative steepness as well as interpret positive versus negative slope.
Calculate the average rate of change between two points on a function.
Determine algebraic expressions for horizontal and vertical lines.
Translate the formulae for lines between slope-intercept, point slope, and general forms.
Determine if two lines are parallel.
Determine if two lines are perpendicular.
Determine the formula for a line that includes a given point and is
parallel or perpendicular to a given line.

Students should be able to do each of the following:

Determine the formula for a line given any set of minimal geometric information. (Example: given two points on the line.)
Express the point slope formula for a line given a written description of a physical situation.
Determine the slopes of parallel and perpendicular lines given written descriptions of a geometric or physical situation.
Construct a linear equation from a written description

Students should be able to do each of the following:

Determine the algebraic formula of a basic function given a written description or a graphical representation.
Determine the algebraic formula for a function after a vertical translation or scaling.
Determine the algebraic formula for a function after a horizontal translation or scaling.
Determine the algebraic formula for a function after a vertical reflection.
Determine the algebraic formula for a function after a combination of vertical and horizontal translations and scalings.
Graph a given function given a set of transformations.
Given the graph of a function, g(x), make sketch of the graph of another function,

\[h(x) = a g(bx+c)+d,\]

where a, b, c, and d are all constants.

Students should be able to do each of the following:

Determine if a simple function is an even or odd function given a graphical representation or algebraic expression.
Define piecewise defined functions with correct notation.
Graph piecewise defined functions.
Identifying the subset of a domain where a function is increasing, decreasing, or is constant.
Determine relative minima and maxima from a graphical representation or for the algebraic representation of simple functions.

Students should be able to do each of the following:

Students should be able to do each of the following:

Determine the algebraic expression for a written description or a graphical representation of a quadratic function.
Graph a quadratic function given a written description or an algebraic expression in any form.
Determine the standard equation for a quadratic function given a written description or a graphical representation.
Determine the vertex of a parabola.
Complete the square to convert a quadratic function into its vertex form.
Determine an optimal solution to a written description of a constrained optimization problem where the resulting cost function can be expressed as a quadratic function.