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College Algebra - MAC 1105 > Functions and Graphs > Flashcards

Functions and Graphs Flashcards

1
Q

Relation

A

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.

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2
Q

Function

A

A relation in which each member of the domain corresponds to exactly one member of the range.

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3
Q

Vertical Line Test for Functions

A

If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

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4
Q

Zeros of a Function

A

The x-values for which f(x) = 0.

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5
Q

Increasing Functions

A
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6
Q

Decreasing Functions

A
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7
Q

Constant Functions

A
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8
Q

Relative Maximum

A
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9
Q

Relative Minimum

A
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10
Q

Symmetry and Tests for Symmetry

A
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11
Q

Even Functions and Their Symmetries

A

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.

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12
Q

Odd Functions and Their Symmetries

A

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.

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13
Q

Identifying Even or Odd Functions from Equations

A
  • 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.
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14
Q

Piecewise Function

A

A function that is defined by two (or more) equations over a specified domain.

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15
Q

Difference Quotient of a Function

A
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16
Q

Slope

A
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17
Q

Positive Slope

A
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18
Q

Negative Slope

A
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19
Q

Zero Slope

A
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20
Q

Undefined Slope

A
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21
Q

Point-Slope Form of the Equation of a Line

A

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 )

22
Q

Slope-Intercept Form of the Equation of a Line

A

The slope-intercept form of the equation of a nonvertical line with slope m and y-intercept b is

y = mx + b

23
Q

Graphing y = mx + b Using the Slope and y-intercept

A
  1. Plot the point containing the y-intercept on the y-axis. This is the point ( 0 , b ).
  2. 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.
  3. 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.
24
Q

Equation of a Horizontal Line

A
25
Q

Equation of a Vertical Line

A
26
Q

General Form of the Equation of a Line

A

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.

27
Q

Using Intercepts to Graph Ax + By + C = 0

A
  1. Find the x-intercept. Let y = 0 and solve for x. Plot the point containing the x-intercept on the x-axis.
  2. Find the y-intercept. Let x = 0 and solve for y. Plot the point containing the y-intercept on the y-axis.
  3. 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.

28
Q

Slope and Parallel Lines

A
  1. If two nonvertical lines are parallel, then they have the same slope.
  2. If two distinct nonvertical lines ahve the same slope, then they are parallel.
  3. The two distinct vertical lines, both with undefined slopes, are parallel.
29
Q

Perpendicular

A

Two lines that intersect at a right angle (90 degrees)

30
Q

Slope and Perpendicular Lines

A
  1. If two nonvertical lines are perpendicular, then the product of their slopes is -1.
  2. If the product of the slopes of two lines is -1, then the lines are perpendicular.
  3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
31
Q

Average Rate of Change of a Function

A
32
Q

Vertical Shifts

A
33
Q

Horizontal Shifts

A
34
Q

Reflection about the x-Axis

A

The graph of y = -f( x ) is the graph of y = f( x ) reflected about the x-axis

35
Q

Reflection of the y-Axis

A

The graph of y = f( -x ) is the graph of y = f( x ) reflected about the y-axis

36
Q

Vertically Stretching and Shrinking Graphs

A
37
Q

Horizontally Stretching and Shrinking Graphs

A
38
Q

Summary of Transformations

A
39
Q

Order of Transformations

A
  1. Horizontal Shifting
  2. Stretching of Shrinking
  3. Reflecting
  4. Vertical Shifting
40
Q

Finding a Function’s Domain

A

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.

41
Q

The Algebra of Functions: Sum, Difference, Product, Quotient

A
42
Q

Composition of Functions

A
43
Q

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

A
44
Q

Definition of the Inverse of a Function

A
45
Q

Finding the Inverse of a Function

A
46
Q

The Horizontal Line Test for Inverse Functions

A
47
Q

The Distance Formula

A
48
Q

The Midpoint Formula

A
49
Q

Definition of a Circle

A
50
Q

Standard Form of the Equation of a Circle

A
51
Q

The General Form of the Equation of a Circle

A

College Algebra - MAC 1105 (6 decks)

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