review math 1113 precalculus fall 2010 instructor: ayona chatterjee

REVIEW

Math 1113Precalculus

Fall 2010Instructor: Ayona Chatterjee

1.1 GRAPHS AND GRAPHING UTILITIES

What to know?

• Graphing points and functions.

• Using the calculator to plot graphs.

• Finding intercepts.• Interpreting graphs.

The graph of an equation in two variables is the set of all points whose coordinates satisfy the equation.

x

y

x y-2 1-1 -20 -31 -22 1

2 3y x

Graphing CalculatorThe Viewing Rectangle

Xmin Xmax

Ymax

Ymin

Xscl

Yscl

The meaning of the terms is demonstrated at right. This is the window that you get with the dimensions chosen by pressing Window in the top row of keys on your calculator.

The Xscl and Yscl are the spacing between tick marks.We express this window’s dimensions as [-8,8,1] by [-4,10,1]. X dimensions are first and Y dimensions second.

To get the standard window [-10,10,1] x [-10,10,1] press Zoom, then the number 6, for Zoom Standard.

Graphing CalculatorMaking a Table

Press 2nd Window in order to Set up the Table.

Press 2nd Graph to see the Table of values.

Press the blue key in the upper left corner that says y= Now type in the equation. X,T,,,,n then X2 then

and finally 9

1.2BASIC FUNCTIONS AND THEIR GRAPHS

What to know?

• What is a function?• What is the domain and range of a function?• The vertical line test.• Finding values of a function at particular

points.• Plotting of functions.

x

y

The first graph is a function, the second is not.

x

y

x

y

Identify the function's domain and range from the graph

Domain (-1,4]

Range [1,3)

Domain [3, )

Range [0, )

x

y

1.3 MORE ON FUNCTIONS AND THEIR GRAPHS

What to know?

• Increasing, decreasing and constant functions.• Relative maxima and minima.• Even and odd functions.• Piecewise functions.• Difference quotient.

Increasing, decreasing and constant functions

The open intervals describing where functions increase, decrease, or are constant, use x-coordinates and not the y-coordinates.

Relative Maxima-Minima

Piecewise function

• A function that is defined over two or more domain is called a piecewise function.

• Example:

f x 3 if - x 3

2 3 if x>3x

1.4 LINEAR FUNCTIONS AND SLOPE

Difference quotient

What to know?

• Slope of a line.• Point-slope form of a line.• Slope intercept form of a line.• Equations of horizontal and vertical lines.• General form of equation of a line.

Slope of a line

Point-slope form of a line

If you are given two points

and you need to write an

equation in point-slope

form, then you can use

either point for (x1,y1).

Slope intercept form of a line

Equations of horizontal and vertical lines

General form of the equation of a line

1.5 MORE ON SLOPE

What to know?

• Slope and parallel lines.• Slope and perpendicular lines.• Average rate of change.

Parallel and Perpendicular lines

Average rate of change

1.6 TRANSFORMATION OF FUNCTIONS

What to know?

• Graphs of common functions.• Vertical shifts.• Horizontal shifts.• Reflection about an axis.• Stretching and shrinking of a graph.• Should be able to transform a graph WITHOUT

a calculator.

x

y

Reciprocal Function

Domain: - ,0 0,

Range: - ,0 0,

Decreasing on - ,0 0,

Odd function

and

1( )f x

x

Summary of transformations

Sequence of transformations

A function involving more than one transformation can be graphed by performing transformations in the following order:

1. Horizontal shifting

2. Stretching or shrinking

3. Reflecting

4. Vertical shifting