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MAT091 :: Lecture Note :: Week 12
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Assignment(s)
To Be Defined...

Review

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Increasing and Decreasing Functions

Functions are sometimes categorized as either increasing or decreasing.

An increasing function is a function whose output values increase (i.e. get larger) when the inputs increase.

   f(0) = 5
   f(1) = 10
   f(2) = 15
   f(3) = 20

   f(x) appears to be an increasing function

A decreasing function is a function whose output values decrease (i.e. get smaller) when the inputs increase.

   g(0) = 100
   g(1) = 90
   g(2) = 80
   g(3) = 70

   g(x) appears to be a decreasing function

A constant function f(x) = k (where 'k' is a constant) is neither increasing nor decreasing.

The identity function f(x) = x is an increasing function.

Some functions can be both increasing and decreasing. For example, over an interval A, a function might be increasing, while it decreases over an interval B. Review the absolute value function.

   abs(x) is decreasing from (-INF, 0) 
   abs(x) is zero at 0
   abs(x) is increasing from (0, INF)

Technical definitions for increasing and decreasing functions.

   increasing function:
   ====================
   Function f(x) increases on an interval I if f(b) > f(a) 
                 for all b > a, where a,b are in I.

   decreasing function:
   ====================
   Function f(x) decreases on an interval I if f(b) < f(a) 
                 for all b > a, where a,b are in I.

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Linear Functions

M-W.com defines linear as follows.

   "of, relating to, resembling, or having a graph that 
    is a line and especially a straight line"

   [...and...]

   "having or being a response or output that is directly 
    proportional to the input"

Observe that the word linear contains the word line.

A linear function is a function is a "first degree polynomial function of one variable. These functions are called 'linear' because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line."

The following are equations for a line.

   slope-intercept form:  y = mx + b
   ... used when slope and y-intercept are known

   point-slope form:  y - y1 = m(x - x1)
   ... where (x1, y1) is a point on the line having slope m

   standard form:  Ax + By = C

Slope is another way of saying "rate of change" and linear functions have a constant rate of change.

Slope is often described as the ratio "rise over run."

Linear functions with a positive constant rate of change are increasing functions. Linear functions with a negative constant rate of change are decreasing functions.

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Slope (Linear Functions)

The slope of a line is a measure of its average rate of change (steepness?). The slope also indicates if the line is increasing (uphill?) or decreasing (downhill?). In linear equations, slope is represented by the letter m.

The following is how to find the slope between two unique points on a line.

   point 1:  (x1, y1)
   point 2:  (x2, y2) 

   slope = m = (y2 - y1) / (x2 - x1)

   note:  x1 ≠ x2

Slope represents the ratio of change-in-output over change-in-input.

   m = Δy / Δx    (change in output / change in input)

   Δ is the "delta" character
   Δy is read "change of" (Δy is read "change of" y)
   Δy = y2 - y1

Slope is often described as the ratio "rise over run."


        rise
   m = ------
        run

Some slope notes.

   horizonal lines have slope 0   (y = f(x) = k, where k is a consant)
   vertical lines have undefined slope   (x = k, where k is a consant)
   parallel lines have equal slopes     (m1 = m2)
   perpendicular lines have negative inverse slopes  (m2 = -1/m1)

The following is a verbose form the slope-intercept equation.

   output = slope * input + initial_value_if_any

   ...or...

   output = average_rate_of_change * input + initial_value_if_any

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Intercepts (Linear Functions)

We frequently need to know the points where lines (graphs in general) intersect (cross) the horizonal-axis and vertical-axis (often called the x-intercept and y-intercept, respectively).

The vertical-intercept for a function is found by evaluating the function when its input is zero. The orderd-pair for a vertical-intercept will be (0, something).

The horizontal-intercept for a function is found by setting its output to zero and finding what input value results in that output. The orderd-pair for a horizonal-intercept will be (something, 0).

   f(x) = 5(x) + 3

   vertical-intercept (y-intercept)... 
      set the input to 0 and evaluate:  f(0) = 5(0) + 3 = 3
      vertical-intercept is (0, 3)

   horizontal-intercept (x-intercept)... 
      set the output to 0 and solve:  0 = 5(x) + 3 
                                     -3 = 5(x)
                                   -3/5 = x
      horizontal-intercept is (-3/5, 0)

Generalizations.

   given  f(x) = m(x) + b

     vertical-intercept:  (0, b)        [notice input is 0]
   horizontal-intercept:  (-b/m, 0)     [notice output is 0]

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Intersecting Lines

Two lines that are not parallel can intersect. If the equations for the lines are known, then the point of intersection can be determined as follows.

   line A:  y = 2x + 2
   line B:  y = 5x + 5

   set the lines equal to one another...
      2x + 2 = 5x + 5

   solve for x...
      2x + 2 + -2 = 5x + 5 + -2    # add -2 to both sides
      2x = 5x + 3

      2x + -5x = 5x + 3 + -5x      # add -5x to both sides
      -3x = 3

      -3x / -3 = 3 / -3            # divide both sides by -3
      x = -1

   substitute for x in one of the line equations and evaluate...
      y = 2(-1) + 2 = -2 + 2 = 0

   point of intersection:  (-1, 0)

   check answer: plug (-1, 0) into the other line equation 
                 and see if it works...
      y = 5(-1) + 5 = -5 + 5 = 0

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Systems of Linear Equations

A "system" of equations is a collection of equations that are treated as an atomic unit. In other words, a system of equations is used when dealing with more than one variable.

   A: 3x + 3y = 0
   B: 12x + 2y = -20

   pick one of the equations...
      pick A

   solve equation A for y
      3x + 3y = 0
      -3x + 3x + 3y = 0 + -3x
      3y = -3x
      3y / 3 = -3x / 3
      y = -x

   substitute for y in the equation B...
      12x + 2(-x) = -20

   solve equation B for x...
      12x + -2x = -20
      10x = -20
      10x / 10 = -20 / 10
      x = -2

   pick one of the equations...
      pick A

   substitute for x in the equation A...
      3(-2) + 3y = 0

   solve equation A for y...
      3(-2) + 3y = 0
      -6 + 3y = 0
      6 + -6 + 3y = 0 + 6
      3y = 6
      3y / 3 =  6 / 3
      y = 2

   the solution is the point (-2, 2)

Another system of linear equations.

   Two numbers have a sum of 55 and a difference of 9. 
   Find the numbers.

   let x and y be the two unknown numbers

   form two equations...
      A: x + y = 55
      B: x - y = 9

   add the two equations together...
         x + y = 55
      +  x - y =  9
      =============
        2x = 64

   solve for x...
      2x = 64
      2x / 2 = 64 / 2
      x = 32

   pick an equation, substitute for x, solve for y
      pick A
      32 + y = 55
      -32 + 32 + y = 55 + -32
      y = 23

   solution:  (32, 23)

   [exercise] Re-do this problem using substitution.

Here is another system of linear equations that is solved using the addition method.

   A: 5x + y = 10
   B:  x + y = 20

   multiply both sides of B by -1...
      -1(x + y) = 20(-1)
      -x - y = -20

   add the two equations...
         5x + y =  10
      + -1x - y = -20
      ===============
         4x     = -10

   solve for x...
      4x / 4 = -10 / 4
       x     = -2.5

   pick an equation and solve for y...
      pick B
      -2.5 + y = 20
      y = 22.5

   solution:  (-2.5, 22.5)

   subtitute the solution into A and see if it works...
      5(-2.5) + 22.5 = 10
      -12.5 + 22.5 = 10
      it works!

Is a system of equations necessary to solve the following problem?

   Ten less than half of a number is thirty. What is the number?

No, because there is only one variable.

   n/2 - 10 = 30
   n/2 - 10 + 10 = 30 + 10
   n/2 = 40
   n/2 * 2 = 40 * 2
   n = 80

Here is a exercise that can be solved using a system of equations.

   A new nightclub had a grand opening in Scottsdale.  First night
   admission was $4 for females and $6 for males.  90 people attended
   the grand opening and the nightclub collected $460.  How many 
   females and how many males attended the grand opening?
   let 'f' represent # of females and let 'm' represent # of males

   A: f + m = 90
   B: $4(f) + $6(m) = $460

   pick A and solve for 'f'...
      f = 90 - m

   substitute 'f' into B...
      4(90 - m) + 6m = 460

   distribute the 4, combine like-terms, solve for 'm'...
      360 - 4m + 6m = 460
      360 + 2m = 460
      -360 + 360 + 2m = 460 - 360
      2m = 100
      (1/2)2m = 100(1/2)
      m = 50

   substitute m = 50 into A and solve for 'f'...
      f + 50 = 90
      f = 40

   There were 40 females and 50 males the grand opening.

Education.Yahoo.com:: Solving Linear Systems

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Solving System of Linear Equations Using TI-83

The "intersection" feature of the TI-83 calculator can be used to find the solution to a system of linear equations.

   press Y=
   Y1 press CLEAR, if necessary
   enter equation:  X is entered using key labeled X,T,theta,n
   -2X + 3 ENTER
   cursor at Y2, press CLEAR if necessary
   enter equation:  
   (1/3)X - 4 ENTER
   press GRAPH
   press 2ND TRACE 5 (intersect)
   First curve? ENTER
   Second curve? ENTER
   Guess? ENTER
   Intersection x = 3 and y = -3   (3, -3)

Substitute x = 3 into both equations and test if (3, -3) is a solution.

   f(x) = -2x + 3
   f(3) = -2(3) + 3 = -6 + 3 = -3     check

   f(x) = (1/3)x - 4
   f(3) = (1/3)(3) - 4 = 1 - 4 = -3   check

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