Welcome to PreCalculus today we will

1
Welcome to Pre-Calculustoday we will

• Review/Discuss Syllabus and Schedule
• Cover sections 2.5 and 4.1
• Cover electronic resources available to assist
  you in the course

2
Chapter 2

• Section 2.5 Piecewise-Defined Functions

3
2.5 Piecewise-Defined Functions

• Piecewise-defined function is a function defined
  by different rules over different subsets of its
  domain.
• Book Example Find each function value given
  the piecewise-defined function
• Solution
• (a)
• (b)
• (c)

4

• Book example continued
• (d) The graph of
• Graph the ray choosing x
  so that with a solid
• endpoint (filled in circle) at (0,2). The
  ray has slope 1 and
• y-intercept 2. Then, graph for
  This graph will be
• half of a parabola with an open endpoint (open
  circle) at (0,0).

Figure 51 pg 2-117
5
2.5 Graphing a Piecewise-Defined Function with
a Graphing Calculator

• Use the test feature
• Returns 1 if true, 0 if false when plotting the
  value of x
• In general, it is best to graph piecewise-defined
  functions in dot mode, especially when the graph
  exhibits discontinuities. Otherwise, the
  calculator may attempt to connect portions of the
  graph that are actually separate from one another.

6
2.5 Graphing a Piecewise-Defined Function

• Sketch the graph of and give f(7) and f(0)
• Solution
• (will do on calculator)
• f(7)
• f(0)
• Now lets try number 20 on page 145 graph by hand
  then check on calculator

7
2.5 The Greatest Integer (Step) Function

• Example Evaluate for (a) 5, (b) 2.46,
  and (c) 6.5
• Solution (a)
• (b)
• (c)
• Using the Graphing Calculator
• The command int is used by many graphing
  calculators for the greatest integer function.

8
2.5 The Graph of the Greatest Integer Function

• Domain
• Range
• If using a graphing calculator, put the
  calculator in dot mode.

Figure 58 pg 2-124
9
2.5 Graphing a Step Function

• Graph the function defined by
  Give the domain and range.
• Solution
• Try some values of x.

10
2.5 Application of a Piecewise-Defined Function

• Downtown Parking charges a 5 base fee for
  parking through 1 hour, and
• 1 for each additional hour or fraction thereof.
  The maximum fee for 24
• hours is 15. Sketch a graph of the function that
  describes this pricing
• scheme.
• Solution
• Sample of ordered pairs (hours,price)
  (.25,5), (.75,5), (1,5), (1.5,6), (1.75,6).
• During the 1st hour price 5
• During the 2nd hour price 6
• During the 3rd hour price 7
• During the 11th hour price 15
• It remains at 15 for the rest of
• the 24-hour period.
• Plot the graph on the interval (0,24.

Figure 62 pg 2-127
11
2.5 Using a Piecewise-Defined Function to
Analyze Data

• Due to acid rain, The percentage of lakes in
  Scandinavia that lost their
• population of brown trout increased dramatically
  between 1940 and 1975.
• Based on a sample of 2850 lakes, this percentage
  can be approximated by
• the piecewise-defined function f .
• Graph f .
• Determine the percentage of lakes that had lost
  brown trout by 1972.

12
2.5 Using a Piecewise-Defined Function to
Analyze Data

• Solution
• (a) Analytic Solution Plot the two endpoints
  and draw the line
• segment of each rule.
• Note Even though there is an open
• circle at the point (1960,18)
• from the first rule, the second
• rule closes it. Therefore, the
• point (1960,18) is closed.

Figure 63 pg 2-128
13
2.5 Using a Piecewise-Defined Function to
Analyze Data

• Graphing Calculator Solution
• (b) Use the second rule with x 1972.

By 1972, about 44 of the lakes had lost their
population of brown trout.
14
Chapter 4

• Section 4.1 Rational Functions and Graphs
• Quick note here is it may be helpful to review
  sections 2.1, 2.2 and 2.3 basics of graphing
  functions and transformations.

15
4.1 Rational Functions and Graphs

• Rational function quotient of two polynomials
• p(x) and q(x), with q(x) ? 0.
• Examples

16
4.1 The Reciprocal Function

• The simplest rational function the reciprocal
  function

17
4.1 The Reciprocal Function

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4.1 Transformations of the Reciprocal Function

• The graph of can be shifted, translated,
  and reflected.
• Example Graph
• Solution The expression
• can be written as
• Stretch vertically by a
• factor of 2 and reflect across
• the y-axis (or x-axis).

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4.1 Graphing a Rational Function

• Example Graph
• Solution Rewrite y
• The graph is shifted left 1 unit and
  stretched
• vertically by a factor of 2.

20
Practice Problem

• Lets try problem 26 page 277

21
4.1 The Rational Function f (x) 1/x2
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4.1 Graphing a Rational Function

• Example Graph
• Solution

Vertical Asymptote x 2 Horizontal
Asymptote y 1.
23
Practice Problem

• Lets try problem 32 page 277

24
4.1 Mode and Window Choices for Calculator
Graphs

• Non-decimal vs. Decimal Window
• A non-decimal window (or connected mode) connects
  plotted points.
• A decimal window (or dot mode) plots points
  without connecting the dots.
• Use a decimal window when plotting rational
  functions such as
• If y is plotted using a non-decimal window, there
  would be a vertical line at x 1, which is not
  part of the graph.

25
4.1 Mode and Window Choices for Calculator
Graphs

• Illustration
• Note See Table for the y-value at x 1 y1
  ERROR.

26
In Pre-Calculustoday we will

• Review/Discuss Homework from 2.5 and 4.1
• Cover section 4.2

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Chapter 4

• Section 4.2 More on Graphs of Rational Functions
• Graphing rational functions beyond transformation
  of the rational
• function or the function
• Graphing rational functions involves several
  stepsthe first of which is graphing asymptotes.

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4.2 More on Graphs of Rational Functions
Vertical and Horizontal Asymptotes

If the function itself (thats the Ys) goes to
either positive or negative infinity (shoots up
or down) at a point along the x-axis then the
value at that point represents a vertical
asymptote x a If the function itself begins to
flatten out to a specific value as x goes to
infinity in either direction (basically it
becomes almost horizontal at y-value) then the
value is a horizontal asymptote y a
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4.2 More on Graphs of Rational Functions

• So how do you find horizontal and vertical
  asymptotes without going straight to the
  calculator and graphing it?
• Vertical is straight forward, set the denominator
  equal to zero and solve for x.the solutions are
  the x values where there are vertical asymptotes
• Horizontal and Oblique asymptotes are a bit more
  complicated there are three possible cases which
  we will cover.
• Numerator lesser degree than denominator
• Numerator and denominator same degree
• Numerator of one higher degree than denominator

30
4.2 Finding Asymptotes Example 1

• Example 1 Find the asymptotes of the graph of
• Solution Vertical asymptotes set denominator
• equal to 0 and solve.

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4.2 Finding Asymptotes Example 1

• Horizontal asymptote divide each term by the
• variable factor of greatest degree, in this case
  x2.
• Therefore, the line y 0 is the horizontal
  asymptote.

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4.2 Finding Asymptotes Example 2

• Example 2 Find the asymptotes of the graph of
• Solution
• Vertical asymptote solve the equation x 3
  0.
• Horizontal asymptote divide each term by x.

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4.2 Finding Asymptotes Example 3

• Example 3 Find the asymptotes of the graph of
• Solution
• Vertical asymptote
• Horizontal asymptote

34
4.2 Finding Asymptotes Example 3

• Rewrite f using synthetic division as follows
• For very large values of is close
  to 0, and
• the graph approaches the line y x 2. This line
  is an
• oblique asymptote (neither vertical nor
  horizontal)
• for the graph of the function.

35
4.2 Determining Asymptotes

• To find asymptotes of a rational function defined
  by a rational
• expression in lowest terms, use the following
  procedures.
• Vertical Asymptotes
• Set the denominator equal to 0 and solve for x.
  If a is a zero of the denominator but not the
  numerator, then the line x a is a vertical
  asymptote.
• Other Asymptotes Consider three possibilities
• If the numerator has lower degree than the
  denominator, there is a horizontal asymptote, y
  0 (x-axis).
• If the numerator and denominator have the same
  degree, and f is

36
4.2 Determining Asymptotes

• Other Asymptotes (continued)
• If the numerator is of degree exactly one greater
  than the denominator, there may be an oblique
  asymptote. To find it, divide the numerator by
  the denominator and disregard any remainder. Set
  the rest of the quotient equal to y to get the
  equation of the asymptote.
• Notes
• The graph of a rational function may have more
  than one vertical asymptote, but can not
  intersect them.
• The graph of a rational function may have only
  one other non-vertical asymptote, and may
  intersect it.

37
4.2 Graphing Rational Functions

• Let define a rational
  expression in lowest terms.
• To sketch its graph, follow these steps.
• Find all asymptotes.
• Find the x- and y-intercepts.
• Determine whether the graph will intersect its
  non-vertical asymptote by solving f (x) k
  where y k is the horizontal asymptote, or f (x)
  mx b where
• y mx b is the equation of the oblique
  asymptote.
• Plot a few selected points, as necessary. Choose
  an x-value between the vertical asymptotes and
  x-intercepts.
• Complete the sketch.

38
4.2 Comprehensive Graph Criteria for a Rational
Function

• A comprehensive graph of a rational function will
  
• exhibits these features
• all intercepts, both x and y
• location of all asymptotes vertical, horizontal,
  and/or oblique
• the point at which the graph intersects its
  non-vertical asymptote (if there is such a
  point)
• enough of the graph to exhibit the correct end
  behavior (i.e. behavior as the graph approaches
  its nonvertical asymptote).

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4.2 Graphing a Rational Function

• Example Graph
• Solution
• Step 1
• Step 2 x-intercept solve f (x) 0

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4.2 Graphing a Rational Function

• y-intercept evaluate f (0)
• Step 3 To determine if the graph intersects the
• horizontal asymptote, solve
• Since the horizontal asymptote is the x-axis,
• the graph intersects it at the point (1,0).

41
4.2 Graphing a Rational Function

• Step 4 Plot a point in each of the intervals
  determined
• by the x-intercepts and vertical asymptotes,
• to get an
• idea of how the graph behaves in each region.
• Step 5 Complete the sketch. The graph approaches
  
• its asymptotes as the points become farther
• away from the origin.

42
4.2 Graphing a Rational Function That Does Not
Intersect Its Horizontal Asymptote

• Example Graph
• Solution Vertical Asymptote
• Horizontal Asymptote
• x-intercept
• y-intercept
• Does the graph intersect the horizontal
  asymptote?

43
4.2 Graphing a Rational Function That Does Not
Intersect Its Horizontal Asymptote

• To complete the graph of
  choose
• points (4,1) and .

44
4.2 Graphing a Rational Function with an
Oblique Asymptote

• Example Graph
• Solution Vertical asymptote
• Oblique asymptote
• x-intercept None since x2 1 has no real
  solutions.
• y-intercept

45
4.2 Graphing a Rational Function with an
Oblique Asymptote

• Does the graph intersect the oblique
  asymptote?
• To complete the graph, choose the points

46
4.2 Graphing a Rational Function with a Hole

• Example Graph
• Solution Notice the domain of the function
  cannot include 2.
• Rewrite f in lowest terms by factoring the
  numerator.

The graph of f is the graph of the line y x
2 with the exception of the point with x-value 2.
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In Pre-Calculustoday we will

• Review/Discuss Homework from previous sections
• Cover section 4.3

48
Chapter 4

• Section 4.3 - Rational Equations, inequalities,
  Applications and Models
• Here we will deal with equations (and
  inequalities) that contain one or more rational
  expressions.
• Again there will be multiple steps to solving the
  rational equation or inequality.
• We will look at equations first, then
  inequalities, applications are mixed.

49
4.3 Solving Equations Involving Rational
Functions

• Determine all values for which the rational
  function is undefined.
• Multiply both sides of the equation by the least
  common denominator of the entire equation.
• Solve the resulting equation.
• Reject any values that were determined in Step 1.
  
• These steps are the analytic (by hand) approach,
  it can also be solved graphically

50
4.3 Solving a Rational Equation Analytically

• Example Solve
• Analytic Solution Notice that the expression is
• undefined for

Multiply both sides by 2x 1.
Solve for x.
The solution set is 1.
51
4.3 Solving a Rational Equation Graphically

• Graphical Solution Rewrite the equation as
• and define Y1 Using
  the
• x-intercept method shows that the zero of the
  function
• is 1.

52
4.3 Solving Inequalities Involving Rational
Functions

• Move all terms to the right side and combine to
  get a single rational expression, a zero will
  remain on the left side.
• Determine values for which the new rational
  function is undefined (denom. Equals zero) or
  zero (numer. Equals zero).
• Use the values from step 2 to develop a truth or
  sign table using test values for each interval.
• Use the results of the table to identify where
  the function satisfies the inequality from step 1
• These steps are the analytic (by hand) approach,
  it can also be solved graphically

53
4.3 Solving a Rational Inequality Analytically

• Example Solve the rational inequality
• Analytic Solution We cant multiply both sides
• by 2x 1 since it may be negative. Start by
• subtracting 1 from both sides.

Common denominator is 2x 1.
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4.3 Solving a Rational Inequality Analytically

• To determine the sign graph, solve the equations
• to get x 1 and

Rewrite as a single fraction.
55
4.3 Solving a Rational Inequality Analytically

• Complete the sign graph and determine the
  intervals
• where the quotient is negative.
• The quotient is zero or negative when x is in
• Cant include it makes the denominator 0.
• The book shows a truth table on page293, either
  is fine.

56
4.3 Solving a Rational Inequality Graphically

• Graphical Solution Let Y1
  We use the
• graph to find the intervals where Y1 is below the
  
• x-axis, including the x-intercepts, where Y1 0.
• The solution set is

57
4.3 Solving a Rational Equation one more
example

• Example Solve
• Solution For this equation,

58
4.3 Solving a Rational Equation

• But, x 2 is not in the domain of the original
• equation and, therefore, must be rejected. The
• solution set is 5.
• Lets Try 38 and 66 on page 302 Analytically and
  check graphically

59
4.3 Applications and Models of Rational
Functions Traffic Intensity

• Example Vehicles arrive randomly at a parking
  ramp at an
• average rate of 2.6 vehicles per minute. The
  parking attendant
• can admit 3.2 cars per minute. However, since
  arrivals are
• random, lines form at various times.
• Traffic intensity x is the ratio of the average
  arrival rate to the average admittance rate.
  Determine x for this parking ramp.
• The average number of vehicles waiting in line to
  enter the ramp is modeled by f (x)
  where 0 ? x lt1 is the traffic intensity. Compute
  f (x) for this parking ramp.
• Graph y f (x). What happens to the number of
  vehicles waiting as the traffic intensity
  approaches 1?
• (This problem is similar to 77 in your
  homework)

60
4.3 Applications and Models of Rational
Functions Traffic Intensity

• Solution
• Average arrival rate 2.6 vehicles/min,
• average admittance rate 3.2 vehicles/min, so
• (b) From part (a), the average number of vehicles
  waiting in line is f (.8125).

61
4.3 Applications and Models of Rational
Functions Traffic Intensity

• (c) From the graph below, we see that as x
  approaches 1, y f (x) gets very large, that
  is, the number of waiting vehicles gets very
  large.

62
4.3 Applications and Models of Rational
Functions Optimization Problem

• Example A manufacturer wants to construct
  cylindrical
• aluminum cans with volume 2000 cm3 (2 liters).
  What radius
• and height will minimize the amount of aluminum
  used?
• What will this amount be?
• Solution Two unknowns radius x
• and height h. To minimize the amount
• of aluminum, we minimize the surface
• area. Volume V is
• (like problem73 on page 303)

63
4.3 Applications and Models of Rational
Functions Optimization Problem

• Surface area S 2?xh 2?x2, x gt 0 (since x is
  the radius), can
• now be written as a function of x.
• Minimum radius is approximately 6.83 cm and the
  height
• associated with that is ?13.65 cm,
  giving a minimum
• amount of aluminum of 878.76 cm3.

64
4.3 Applications and Models of Rational
Functions Variation

• There are several types of variation
• Direct y varies directly as x
• Inverse y varies inversely as x
• Jointly y varies jointly as x and z
• All three forms may also include a power i.e. y
  varies directly as the second power of x or y is
  inversely proportional to x cubed

65
4.3 Modeling the Intensity of Light

• The intensity of light, I is inversely
  proportional to the second power of the distance
  d. The equation
• models this phenomenon. At a distance of 3
  meters, a 100-watt bulb produces an intensity of
  .88 watt per square meter. Find the constant of
  variation k, and then determine the intensity of
  the light at a distance of 2 meters.
• Substitute d 3, and I .88 into the variation
  equation, and solve for k.

66
4.3 Solving a Combined Variation Problem in
Photography

• In the photography formula
• the luminance L (in foot-candles) varies directly
  as the square of the F-stop F and inversely as
  the product of the file ASA number s and the
  shutter speed t. The constant of variation is 25.
• Suppose we want to use 200 ASA file and a shutter
  speed of 1/250 when 500 foot candles of light are
  available. What would be an appropriate F-stop?

An F-stop of 4 would be appropriate.
67
In Pre-Calculustoday we will

• Cover section 4.4 and 4.5
• Review/Discuss Homework from previous sections

68
Chapter 4

• Section 4.4 Functions Defined by Powers and
  Roots

69
4.4 Functions Defined by Powers and Roots

• f (x) xp/q, p/q in lowest terms
• if q is odd, the domain is all real numbers
• if q is even, the domain is all nonnegative real
  numbers

Power and Root Functions A function f given by
f (x) xb, where b is a constant, is a power
function. If , for some integer n ? 2,
then f is a root function given by f (x)
x1/n, or equivalently, f (x)
70
4.4 Graphing Power Functions

• Example Graph f (x) xb, b .3, 1, and 1.7,
  for
• x ? 0.
• Solution The larger values of b cause the graph
  of
• f to increase faster.

71
4.4 Modeling Wing Size of a Bird

• Example Heavier birds have larger wings with
  more surface
• area. For some species of birds, this
  relationship can be
• modeled by S (x) .2x2/3, where x is the weight
  of the bird in
• kilograms and S is the surface area of the wings
  in square
• meters. Approximate S(.5) and interpret the
  result.
• Solution
• The wings of a bird that weighs .5 kilogram have
  a surface
• area of about .126 square meter.

72
4.4 Modeling the Length of a Birds Wing

• Example The table lists the weight W and the
• wingspan L for birds of a particular species.
• Use power regression to model the data with
• L aWb. Graph the data and the equation.
• (b) Approximate the wingspan for a bird weighing
  3.2 kilograms.

W (in kilograms)
L (in meters)
73
4.4 Modeling the Length of a Birds Wing

• Solution
• (a) Let x be the weight W and y be the length L.
  Enter the data, and then select power regression
• (PwrReg), as shown in the following figures.

74
4.4 Modeling the Length of a Birds Wing

• The resulting equation and graph can be seen in
  the figures below.
• (b) If a bird weighs 3.2 kg, this model predicts
  the wingspan to be

75
4.4 Graphs of Root Functions Even Roots

76
4.4 Graphs of Root Functions Odd Roots

77
4.4 Finding Domains of Root Functions

• Example Find the domain of each function.
• (a) (b)
• Solution
• 4x 12 must be greater than or equal to 0 since
  the root, n 2, is even.
• (b) Since the root, n 3, is odd, the domain of
  g is all real numbers.

The domain of f is 3,?).
78
4.4 Transforming Graphs of Root Functions

• Example Explain how the graph of
• can be obtained from the graph of
• Solution

Shift left 3 units and stretch
vertically by a factor of 2.
79
4.4 Transforming Graphs of Root Functions

• Example Explain how the graph of
• can be obtained from the graph of
• Solution

Shift right 1 unit, stretch
vertically by a factor of 2, and reflect across
the x-axis.
80
4.4 Graphing Circles Using Root Functions

• The equation of a circle centered at the origin
  with radius r is found by finding the distance
  from the origin to a point (x,y) on the circle.
• The circle is not a function, so imagine a
  semicircle on top and another on the bottom.

81
4.4 Graphing Circles Using Root Functions

• Solve for y
• Since y2 y1, the bottom semicircle is a
  reflection of the top semicircle.

82
4.4 Graphing a Circle

• Example Use a calculator in function mode to
• graph the circle
• Solution This graph can be obtained by graphing
• in the same
• window.

Technology Note Graphs may not connect when
using a non-decimal window.
83
4.5 Equations, Inequalities, and Applications
Involving Root Functions

• Note This property does not say that the two
• equations are equivalent. The new equation may
• have more solutions than the original.
• e.g.

Power Property If P and Q are algebraic
expressions, then every solution of the equation
P Q is also a solution of the equation Pn Qn,
for any positive integer n.
84
4.5 Solving Equations Involving Root Functions

• Isolate a term involving a root on one side of
  the equation.
• Raise both sides of the equation to a power that
  will eliminate the radical or rational exponent.
• Solve the resulting equation. (If a root is still
  present after Step 2, repeat Steps 1 and 2.)
• Check each proposed solution in the original
  equation.

85
4.5 Solving an Equation Involving Square Roots

• Example Solve
• Analytic Solution

Isolate the radical.
Square both sides.
Write in standard form and solve.
86
4.5 Solving an Equation Involving Square Roots

• These solutions must be checked in the original
• equation.

87
4.5 Solving an Equation Involving Square Roots

• Graphical Solution The equation in the second
  step
• of the analytic solution has the same solution
  set as
• the original equation. Graph
• and solve y1 y2.
• The only solution is at x 2.

88
4.5 Solving an Equation Involving Cube Roots

• Example Solve
• Solution

89
4.5 Solving an Equation Involving Roots
(Squaring Twice)

• Example Solve
• Solution

Isolate radical.
Square both sides.
Isolate radical.
Square both sides.
Write in standard quadratic form and solve.
A check shows that 1 and 3 are solutions of the
original equation.
90
4.5 Solving Inequalities Involving Rational
Exponents

• Example Solve the inequality
• Solution The associated equation solution in
  the
• previous example was
• Let

Use the x-intercept method to solve this
inequality and determine the interval where the
graph lies below the x-axis. The solution is the
interval .
91
4.5 Application Solving a Cable Installation
Problem

• A company wishes to run a utility cable from
  point A on the
• shore to an installation at point B on the island
  (see figure).
• The island is 6 miles from shore. It costs 400
  per mile to run
• cable on land and 500 per mile underwater.
  Assume that the
• cable starts at point A and runs along the
  shoreline, then
• angles and runs underwater
• to the island. Let x represent
• the distance from C at which
• the underwater portion of the
• cable run begins, and the
• distance between A and C be
• 9 miles.

92
4.5 Application Solving a Cable Installation
Problem

• What are the possible values of x in this
  problem?
• Express the cost of laying the cable as a
  function of x.
• Find the total cost if three miles of cable are
  on land.
• Find the point at which the line should begin to
  angle in order to minimize the total cost. What
  is this total cost?
• Solution
• The value of x must be real where
• Let k be the length underwater. Using the
  Pythagorean theorem,

93
4.5 Application Solving a Cable Installation
Problem

• The cost of running cable is price ? miles. If
  C is the total cost (in dollars) of laying cable
  across land and underwater, then
• If 3 miles of cable are on land, then 3 9 x,
  giving
• x 6.

94
4.5 Application Solving a Cable Installation
Problem

• Using the graphing calculator, find the minimum
  value of y1 C(x) on the interval (0,9.
• The minimum value of the function occurs when x
  8. So 9 8 1 mile should be along land, and
• miles underwater. The cost is