Functions and Graphs Flashcards
Relation
Any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation.
Function
A relation in which each member of the domain corresponds to exactly one member of the range.
Vertical Line Test for Functions
If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.
Zeros of a Function
The x-values for which f(x) = 0.
Increasing Functions

Decreasing Functions

Constant Functions

Relative Maximum

Relative Minimum

Symmetry and Tests for Symmetry

Even Functions and Their Symmetries
The function f is an even function if
f( -x ) = f( x ) for all x in the domain of f.
The graph of an even function is symmetric with respect to the y-axis.
Odd Functions and Their Symmetries
The function f is an even function if
f( -x ) = - f( x ) for all x in the domain of f.
The graph of an even function is symmetric with respect to the origin.
Identifying Even or Odd Functions from Equations
- Even function: f( -x ) = f( x )
The right side of the equation of an even function does not change if x is replaced with -x. - Odd function: f( -x ) = -f( x )
Every term on the right side of the equation of an odd function changes sign if x is replaced with -x. - Neither even nor odd: f( -x ) <> f( x ) and f( -x ) <> -f( x )
The right side of the equation changes if x is replaced with -x, but not every term on the right side changes sign.
Piecewise Function
A function that is defined by two (or more) equations over a specified domain.
Difference Quotient of a Function

Slope

Positive Slope

Negative Slope

Zero Slope

Undefined Slope

Point-Slope Form of the Equation of a Line
The point-slope form of the equation of a nonvertical line with slope m that passed through the point ( x1 , y1 ) is
y - y1 = m ( x - x1 )
Slope-Intercept Form of the Equation of a Line
The slope-intercept form of the equation of a nonvertical line with slope m and y-intercept b is
y = mx + b
Graphing y = mx + b Using the Slope and y-intercept
- Plot the point containing the y-intercept on the y-axis. This is the point ( 0 , b ).
- Opbtain a second point using the slope, m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point.
- Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of teh line to show that the line continues indefinitely in both directions.
Equation of a Horizontal Line

Equation of a Vertical Line

General Form of the Equation of a Line
Every Line has an equation that can be written in the general form
Ax + By + C = 0,
where A, B, and C are real numbers, and A and B are not both zero.
Using Intercepts to Graph Ax + By + C = 0
- Find the x-intercept. Let y = 0 and solve for x. Plot the point containing the x-intercept on the x-axis.
- Find the y-intercept. Let x = 0 and solve for y. Plot the point containing the y-intercept on the y-axis.
- Use a straightedge to draw a line through the two points containing the intercepts. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.
As long as none of A, B, and C is zero, the graph of Ax + By + C = 0 will have distinct x- and y-intercepts, and this three step method an be used to graph the equation.
Slope and Parallel Lines
- If two nonvertical lines are parallel, then they have the same slope.
- If two distinct nonvertical lines ahve the same slope, then they are parallel.
- The two distinct vertical lines, both with undefined slopes, are parallel.
Perpendicular
Two lines that intersect at a right angle (90 degrees)
Slope and Perpendicular Lines
- If two nonvertical lines are perpendicular, then the product of their slopes is -1.
- If the product of the slopes of two lines is -1, then the lines are perpendicular.
- A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
Average Rate of Change of a Function

Vertical Shifts

Horizontal Shifts

Reflection about the x-Axis
The graph of y = -f( x ) is the graph of y = f( x ) reflected about the x-axis
Reflection of the y-Axis
The graph of y = f( -x ) is the graph of y = f( x ) reflected about the y-axis
Vertically Stretching and Shrinking Graphs

Horizontally Stretching and Shrinking Graphs

Summary of Transformations

Order of Transformations
- Horizontal Shifting
- Stretching of Shrinking
- Reflecting
- Vertical Shifting
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f( x ) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in an even rood, such as a square root, of a negative number.
The Algebra of Functions: Sum, Difference, Product, Quotient

Composition of Functions

Excluding Values from the Domain of (f o g)( x ) = f( g ( x ) )

Definition of the Inverse of a Function

Finding the Inverse of a Function

The Horizontal Line Test for Inverse Functions

The Distance Formula

The Midpoint Formula

Definition of a Circle

Standard Form of the Equation of a Circle

The General Form of the Equation of a Circle
