Linear Models

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Chapter 3

• Section 1
• Linear Models

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Homework 9/12

• Assignment 7 (due Tuesday, Sept. 26)

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Linear Models

• Myra is saving money to buy a next X-BOX 360 She
  estimates shell need 950 and that she can
  reasonably save 60 per month. The amount of
  money saved is dependent on the months. So, well
  let months t and label our axes as shown.

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• Some things to note
• The horizontal and vertical axes have different
  scales.
• The axes are labeled and the appropriate units
  are included

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• On our graph, plot the following
• At month 0, Myras parents gave her 600 towards
  the X-BOX.
• After 1 month she had how much money saved? Plot
  this point,
• Continue in this fashion until Myra has saved
  enough money to buy the X-BOX.
• How many months did it take her?
• Is this a discrete or continuous model of the
  relationship between time and money?

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The Model Equation

• Myra saves at a rate of 60/month
• She started with 600
• Make a table of values.
• Write an equation that models this situation.

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Discrete vs Continuous

• If we wanted to model the number of eggs coming
  out of a hen house over a weeks period of time we
  would use a discrete model.
• If we wanted to model the amount of milk coming
  from a heard over a weeks time we would use a
  continuous model.

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• Please get out a piece of graph paper.
• Temperature is usually measured in Fahrenheit in
  the US while the rest of the world uses Celsius.
  Lets model the relationship between these two
  systems by letting C be our independent variable
  and F be our dependent. Note that 0C is the same
  as 32F and the 100C is 212F. The relation is
  linear. So, label and scale a coordinate system
  and plot these 2 point.

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• At what rate is Fahrenheit changing with respect
  to Celsius?
• What is the initial or starting amount?
• Make a table of values
• Write the model

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• According to the Statistical Abstract of the US
  there were approximately 31,000 crimes reported
  in the US in 1998, and this amount was dropping
  by a rate of about 2900 per year.
• Let C be the crimes committed x years after 1998.
  So, C(0) 31,000. Explain what this means.
• What is the model equation that compares crimes
  to years after 1998?
• Use your model to predict the number of crimes
  committed in 2003.

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Chapter 3

• Section 2
• Slope

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Definition

• The Slope of a line describes the rate at which
  the line rises or falls. It is a measure of the
  steepness of the line.
• The Slope of a line is the rate at which the
  dependent variable is changing with respect to
  the independent variable.

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Example

• Sean rides up Korbel hill and wants to know how
  steep it is. Using his GPS unit he gets the
  following numbers. At the base of the hill he is
  at 50 feet above sea level. After moving 100 feet
  horizontally he is 60 feet above sea level.

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How Steep?
(100,60)
(0,50)
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The Slope Formula
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Example

• Plot the following points, then calculate the
  slope(a) by drawing the triangle between the two
  points(b) by using the slope formula

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• Compute the slope of the given line.

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Parallel Lines

• If two lines have the same slope then they are
  parallel.

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Example

• Draw the line through (1, 2) having slope 1/3.
• On the same coordinate system draw the line
  through (0, 1) having slope 1/3.

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Perpendicular Lines
Rotate line RS 90
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Perpendicular Lines

• If the slope of a given line is m1 then a
  perpendicular line will have slope given by m2
  such that

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Example

• Draw the line through (1, 2) having slope 1/3.
• On the same coordinate system draw the line
  through (0, 1) having slope 3.