7'1 Review of Graphs and Slopes of Lines - PowerPoint PPT Presentation

1 / 35
About This Presentation
Title:

7'1 Review of Graphs and Slopes of Lines

Description:

The graph of any linear equation in two variables is a straight line. ... 9.2 Absolute Value Inequalities - Form 1. Solving equations of the form: Setup the compound ... – PowerPoint PPT presentation

Number of Views:22
Avg rating:3.0/5.0
Slides: 36
Provided by: davidh98
Category:
Tags: form1 | graphs | lines | review | slopes

less

Transcript and Presenter's Notes

Title: 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
Write a Comment
User Comments (0)
About PowerShow.com