Graphing and Writing Linear Equations

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Chapter 5

• Graphing and Writing Linear Equations

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Warm Up 5-1

• What do you think Slope means?
• Brainstorm a list of things that vary in slope.
  Describe ways in which the steepness of a slope
  might be measured?

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5-1 Slope

• OBJ Calculate the slope of a line
• Draw a line through a point with a given
  slope.

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• Carpenters use the terms rise and run to describe
  the steepness of a stairway or a roof line.
• You can use rise and run to describe the
  steepness of a hill
• Steepness Rise/Run

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Definition

• Remember slope is Rise/Run

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Examples

• Find the slope of the line through (3,1) and
  (-5,7)
• Find the slope of the line through (2,1) and
  (6,7)
• If a line goes up 5 units for each 2 units that
  it goes to the right, what is the slope of that
  line.

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• Show that (1,5)(3,11) and (-3,-7) lie on the same
  line. Find the slope of that line.
• Find the slope of the line with equation x-4y 8
• Find the slope of the line with equation 3x 4y 6

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• Draw a line through the point (1,2) with the
  slope -3/2
• What are the coordinates of the second point in
  this example?
• Graph the line through point (2,3) with the slope
  5/4

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Slope of Lines

• A Line with a positive slope goes upward from
  left to right

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• A line with a negative slope goes downward from
  left to right

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• The slope of a horizontal line is 0

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• The slope of a vertical line is undefined

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5-1 Wrap Up

• What is slope? How do you describe the steepness
  of a slope?

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Warm Up 5-2

• What terms have you used in the past to describe
  how quickly or how slowly something changes?

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5-2 Rates of Change

• OBJ Finding rates of change from table and
  graphs
• Find rates of change in real world situations

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What is Rate of Change?

• Change between two things
• It measures the amount of change in a situation
• Change in the dependent variable/ change in
  independent variable

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Examples

• Suppose you type 140 words in 4 minutes. What is
  your typing rate?

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Definition of Rate of Change
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Relationships between rates of change
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Example Using a Table

• A car rental company charges its customers a base
  relate of 50/day. In addition, the rental
  company charges customers 30 for the first 100
  miles and an additional fee for every mile over
  100 they drive. Find the rate of change charged
  for extra mileage.

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• Laura wished to measure the effectiveness of her
  new exercise program. At the end of every week,
  she counts the number of sit ups that she can do
  in one minute. If she does 30 sit ups the first
  week, 42 sit ups the second week, and 54 sit ups
  the third week, is her progress linear?

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5-2 Wrap Up

• How could you describe a rate of change using a
  table or a graph?

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5-3 Warm Up

• Describe patterns that contain quantities that
  double. For example, the number of shoes in five
  pairs or the number of wings on eight birds.
  Express these patterns as equations with two
  variables

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5-3 Direct Variation

• OBJ Relating slope to constant of variation
• Using constant of variation to solve problems
• Solve real world problems

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Lets talk about deposits

• R 10c
• We say that r varies directly as c

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Direct-Variation Functions

• R 10c is in the form y kx
• where k is a non-zero constant,
• the constant of variation

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Definition

• Direct-variation function is a function with a
  formula of the form ykx, with k not equal to
  zero.
• It is a linear function
• Direct variation functions will always pass
  through the origin (0,0)

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Consider ykx

• You can write this as ky/x
• K is the constant of variation
• K is the rate of change for the data that
  describes the variation
• The value of k is also the slope of the line

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Decide if each equation is a direct variation. If
it is, find the constant of variation

• 2x-3y1
• 2x-3y0
• 5x2y0
• 5x2y9
• 7y2x
• 3y4x8
• Y-7.5x0

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Examples

• The weight that an object exerts on a scale
  varies directly with the mass of the object. If a
  bowling ball has a mass of 6kg, the scale reads
  59 N. Write an equation for the relationship
  between weight and mass.

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• The time it takes you to hear thunder varies
  directly with your distance form the lighting. If
  you are 2 mi from where lightning strikes, you
  will hear thunder about 10 s after you see the
  lightning. Write an equation for the relationship
  between time and distance. About how far are you
  from lighting if you hear thunder 7s after seeing
  lightning?

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Write a function rule for the relationship
between the data
Is this an example of direct variation? Why or
Why not?
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• The force you must apply to lift an object varies
  directly with the objects weight. You need to
  apply 0.625 lb of force to the windlass to lift a
  28 lb weight. How much force would you need to
  lift 100 lb?

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5-3 Wrap Up

• What is direct variation? What relationship does
  direct variation describe?

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Warm Up 5-4

• Sketch a line passing through (0,2). Compare your
  lines with people around you. Did everyone draw
  the same line? How does your line differ?

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5-4 Slope-Intercept Form

• OBJ Using the slope and y-intercept to draw
  graph and write equations

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The y-intercept

• The y intercept is the point on the line where it
  cross the y axis
• (0,b).

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Slope- Intercept Form of a Line

• Y mx b where m is the slope and b is the y
  intercept

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Examples ( give y intercept and slope)

• Y -6x 8
• Y-2x 6
• Y164x
• Y1/2 x 5
• 3x4y 9
• 5x 3y 10

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• Write an equation for the line with slope 6 and y
  intercept of 1. Graph that line.
• Write an equation for the line with y intercept 8
  and has slope -5

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• Jeremy had 42 in his savings account when he
  decided to start adding 5 a week. Write an
  equation for Jeremy's total savings y after x
  weeks. Give the slope and the y intercept.

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Graph the equations

• Y -1/2x 3
• Y 3x-1
• Y-3/2x2

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• The base pay of a water delivery person is 20
  per week. He can also earn 20 commission on any
  sales he makes. What is an equation that models
  this situation. Rewrite the equation in slope
  intercept form. Then graph the equation.

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Wrap Up 5-4

• Can you explain how changing the values of m and
  b affect the graph of the equation ymxb

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Warm Up 5-5

• Your friend call you one night with a question
  about her algebra homework. You decide that the
  best way to help her is to have her draw a line
  of an equation on a graph. How would you describe
  the line without actually showing it to her?

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5-5 Writing an Equation of a Line

• OBJ Write an equation give the slope and a point
• Write an equation give two points from a
  graph or table.

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How to get an Equation with slope and point

• Substitute for m, x, and y
• Solve for b
• Plug in for m and b
• 4. Finish

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Examples

• Find an equation in slope-intercept form for the
  line through (-3,5) with a slope of 2
• A line has slope 4 and passes through point (5,3)
  what is the equation.

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What is the x-intercept

• The point the graph passes the x axis. The
  coordinates are (x, 0)

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• A line has slope 4 and its x intercept is 6,
  find the equation of the line.
• Find the equation of a line that passes through
  (-4, 11) with slope 1
• Find equation of the line through (6, -5) and
  slope 1/2

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• The population of the province of Ontario in
  Canada was 10,085,000 in 1991. At that time, the
  population was increasing at a rate of about
  80,000 people per year. Assume this rates
  continues. Find an equation relating the
  population of Ontario to the year. Use the
  equation to predict the population of Ontario in
  2009

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How to Find Equation through two points

• Calculate slope with given points
• Plug in for x, y, and m
• Solve for b
• Plug in for m and y
• Finish

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Examples

• Find an equation for the line through (5, -1) and
  (-3,3)
• Find an equation for the line through (8,7) and
  (-4,-2)

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• A printer charges 65 for 100 copies of a booklet
  and 105 for 500 copies. Assume the relationships
  between the number of copies and the cost is
  linear. Write an equation relating the cost and
  the number of copies. How many copies can be
  printed for 200

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History Note

• This method of finding an equation for a line
  through two points was developed by Rene
  Descartes in the early 1600s

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Wrap Up 5-5

• Given two points on a line, how do you find the
  equation of that line?

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5-6 Warm Up

• What do you think a scatter plot is? Do you think
  there is a line that can describe the trend of a
  scatter plot?

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5-6 Scatter Plots and Equations of Lines

• OBJ Finding the equation of a trend line
• Find the line of best fit

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Trend Line

• Sometimes you can describe data that show a
  positive or negative correlation with a trend
  line. Then you can use the trend lien to make
  prediction

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How to find a trend line

• Make a scatter plot of the data
• Draw a line that best fits the data (This line
  does not have to go through or touch every point,
  in fact in can not even touch any points)
• Pick two points on your line
• Find the equation of your line

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• A film usually makes the most money in ticket
  sales during the first few weeks after its
  release. Find the equation of a trend line for
  the data about the tickets sales for Forrest Gump

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• Predict ticket sales for the film in its tenth
  week of release.

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Line of Best Fit

• The most accurate trend line is the line of best
  fit
• A graphing calculator can find the line of best
  fit by calculating the correlation coefficient r
• The closer r is to 1 the better fit the line.
• Lets use the calculators for example 2 page 242

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Wrap Up 5-6

• What is a trend line?
• What information can you get from a trend line?
• How is a trend line similar to and different from
  a line of best fit?

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Warm Up 5-7

• How is the slope intercept form of an equation
  like or unlike a recipe?

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5-7 AxByC Form

• OBJ Graph equations using x and y intercepts
• Write equations in AxByC form Model
  situations with equations in the form AxByC

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• The Baker family bought 4 sandwiches and 3
  salads. They spent 24.00. If x is the cost of a
  sandwich and y the cost of a salad, then 4x3y24

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Standard Form of an Equation for a Line

• The equation 4x3y24 is the standard form of an
  equation for a line
• AxBy C

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Examples

• Graph 5x-2y-20
• Graph 4x2y6
• Hamburger sells for 6 pound. Steak cost 12 a
  pound. Chris has 96 dollars to buy hamburger and
  steaks. Write an equation in standard form to
  describe the different possible combinations.

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Examples

• Rewrite y 3/8x ½ in standard form using
  integer values for A,B and C.
• Rewrite each equation in standard form using
  integer values for A, B, and C
• Y-7x-2 x3/4y
• .4x.75y5

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5-7 Wrap Up

• Why is it useful to have two forms of a linear
  equation?

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Warm Up 5-8

• What is a perpendicular line? What math language
  have you learned that you could use to describe a
  line that is perpendicular to a line you just
  graphed?

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5-8 Parallel and Perpendicular Lines

• OBJ Write equations for parallel and
  perpendicular lines
• Use slope to determine if lines are parallel,
  perpendicular or neither

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Parallel Lines

• Two lines are parallel if they have the same
  slope and different y intercepts
• Are horizontal lines parallel? Explain?
• What is the slope of a line parallel to y 3/5 x
  -4?

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Examples

• Write an equation for a line that contains (-2,3)
  and is parallel to the graph of 5x-2y8
• Write an equation for a line that contains
  (-2,-1) and is parallel to the graph of -3x2y -3

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Perpendicular Lines

• Two Lines are Perpendicular if you multiply their
  slopes and get -1
• Find the slope of a line perpendicular to a line
  with the following slopes a. -2 b. 2/7 c. 1/5
  d. 0

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Example

• A sketch of a roof is drawn with one side
  represented by the equation 7x3y4. Write the
  equation for the line representing the other side
  fo the roof which goes through the origin and is
  perpendicular to the other side.

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Wrap Up 5-8

• What are all the differences you can name between
  perpendicular and parallel lines?