7'1 Review of Graphs and Slopes of Lines

1
7.1 Review of Graphs and Slopes of Lines

• Standard form of a linear equation
• The graph of any linear equation in two variables
  is a straight line. Note Two points determine a
  line.
• Graphing a linear equation
• Plot 3 or more points (the third point is used as
  a check of your calculation)
• Connect the points with a straight line.

2
7.1 Review of Graphs and Slopes of Lines

• Finding the x-intercept (where the line crosses
  the x-axis) let y0 and solve for x
• Finding the y-intercept (where the line crosses
  the y-axis) let x0 and solve for yNote the
  intercepts may be used to graph the line.

3
7.1 Review of Graphs and Slopes of Lines

• If y k, then the graph is a horizontal line
  (slope 0)
• If x k, then the graph is a vertical line
  (slope undefined)

4
7.1 Review of Graphs and Slopes of Lines

• Slope of a line through points (x1, y1) and (x2,
  y2) is
• Positive slope rises from left to
  right.Negative slope falls from left to right

5
7.1 Review of Graphs and Slopes of Lines

• Using the slope and a point to graph linesGraph
  the line with slope passing through the point
  (0, 0)Go over 5 (run) and up 3 (rise) to get
  point (5, 3) and draw a line through both
  points.

6
7.1 Review of Graphs and Slopes of Lines

• Finding the slope of a line from its equation
• Solve the equation for y
• The slope is given by the coefficient of x
• Parallel and perpendicular lines
• Parallel lines have the same slope
• Perpendicular lines have slopes that are negative
  reciprocals of each other

7
7.1 Review of Graphs and Slopes of Lines

• Example Decide whether the lines are parallel,
  perpendicular, or neither
• solving for yin first equation
• solving for yin second equation
• The slopes are negative reciprocals of each other
  so the lines are perpendicular

8
7.2 Review of Equations of Lines

• Standard form
• Slope-intercept form(where m slope and b
  y-intercept)
• Point-slope form The line with slope m going
  through point (x1, y1) has the equation

9
7.2 Review of Equations of Lines

• Example Find the equation in slope-intercept
  form of a line passing through the point (-4,5)
  and perpendicular to the line 2x 3y 6
• solve for y to get slope of line
• take the negative reciprocal to get the ? slope

10
7.2 Review of Equations of Lines

• Example (continued)
• Use the point-slope form with this slope and the
  point (-4,5)
• Add 5 to both sides to get in slope intercept
  form

11
7.3 Functions Relations

• Relation Set of ordered pairsExample R
  (1, 2), (3, 4), (5, 1)
• Domain Set of all possible x-values
• Range Set of all possible y-values
• What is the domain of the relation R?

12
7.3 FunctionsRelations
Rangey-values (output)
Domainx-values (input)
Example Demand for a product depends on its
price.Question If a price could produce more
than one demand would the relation be useful?
13
7.3 Functions - Determining Whether a Relation or
Graph is a Function

• A relation is a function if for each x-value
  there is exactly one y-value
• Function (1, 1), (3, 9), (5, 25)
• Not a function (1, 1), (1, 2), (1, 3)
• Vertical Line Test if any vertical line
  intersects the graph in more than one point, then
  the graph does not represent a function

14
7.3 Functions

• Function notation y f(x) read y
  equals f of xnote this is not f times x
• Linear function f(x) mx bExample f(x)
  5x 3
• What is f(2)?

15
7.3 Functions - Graph of a Function

• Graph of
• Does this pass the vertical line test?What is
  the domain and the range?

16
7.3 Functions - Graph of a Parabola
Vertex
17
7.4 Variation

• Types of variation
• y varies directly as x
• y varies directly as the nth power of x
• y varies inversely as x
• y varies inversely as the nth power of x

18
7.4 Variation

• Solving a variation problem
• Write the variation equation.
• Substitute the initial values and solve for k.
• Rewrite the variation equation with the value of
  k from step 2.
• Solve the problem using this equation.

19
7.4 Variation

• Example If t varies inversely as s and t 3
  when s 5, find s when t 5
• Give the equation
• Solve for k
• Plug in k 15
• When t 5

20
9.2 Review Things to Remember

• Multiplying/dividing by a negative number
  reverses the sign of the inequality
• The inequality y gt x is the same as x lt y
• Interval Notation
• Use a square bracket when the endpoint is
  included
• Use a round parenthesis ( when the endpoint is
  not included
• Use round parenthesis for infinity (?)

21
9.2 Review - Compound Inequalities and Interval
Notation

• Solve eachinequality for xTake the
  intersection(why does the order
  change?)Express in interval notation

1
3
22
9.2 Review - Compound Inequalities and Interval
Notation

• Solve eachinequalityfor xTake the
  unionExpress ininterval notation

-1
23
9.2 Review - Absolute Value Equations

• Solving equations of the form

24
9.2 Absolute Value Inequalities

• To solve where k gt 0, solve
  the compound inequality (intersection)
• To solve where k gt 0, solve
  the compound inequality (union)Why cant you
  say ?

25
9.2 A Picture of What is Happening
y

• Graphs ofand f(x) kThe part below the
  line f(x) k is where The part above the line
  f(x) k is where

f(x) k
x
26
9.2 Absolute Value Inequalities - Form 1

• Solving equations of the form
• Setup the compoundinequality
• Subtract 4 all the wayacross
• Divide by 3
• Put into intervalnotation

27
9.2 Absolute Value Inequalities - Form 2

• Solving equations of the form
• Setup the compoundinequality
• Subtract 4 all the wayacross
• Divide by 3
• Put into interval notationWhat part of the real
  line is missing?

28
9.2 Absolute Value Inequalitythat involves
rewriting

• ExampleAdd 3 to both sides (why?)Set up
  compound equationAdd 2 all the way
  acrossPut into interval notation

29
9.2 Absolute Value Inequalities

• Special case 1 when k lt 0Since absolute value
  expressions can never be negative, there is no
  solution to this inequality. In set notation

30
9.2 Absolute Value Inequalities

• Special case 2 when k 0Since absolute value
  expressions can never be negative, there is one
  solution for thisIn set notation What if
  the inequality were lt?

31
9.2 Absolute Value Inequalities

• Special case 3Since absolute value expressions
  are always greater than or equal to zero, the
  solution set is all real numbers. In interval
  notation

32
9.2 A Picture of What HappensWhen k is Negative
y

• Graphs ofand f(x) k
  never gets below the line f(x) k so there is
  no solution toand the solution to
  is all real numbers

x
f(x) k
33
9.2 Relative Error

• Absolute value is used to find the relative error
  of a measurement. If xt represents the expected
  value of a measurement and x represents the
  actual measurement, thenrelative error in

34
9.2 Example of Relative Error

• A machine filling quart milk cartons is set for a
  relative error no greater than .05. In this
  example, xt 32 oz. soSolving this
  inequality for x gives a range of values for
  carton size within the relative error
  specification.

35
9.2 Solution to the Example

• Simplify
• Change into acompound inequality
• Subtract 1
• Multiply by 32
• Reverse the inequality
• Put into interval notation