FUNCTIONS AND GRAPHS

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FUNCTIONS AND GRAPHS
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Aim 1.2 What are the basics of functions and
their graphs?

• Lets Review
• What is the Cartesian Plane or Rectangular
  Coordinate Plans?
• How do we find the x and y-intercepts of any
  function?
• How do we interpret the viewing rectangle
• -10,10, 1 by -10, 10,1?

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IN THIS SECTION WE WILL LEARN

• How to find the domain and range?
• Determine whether a relation is a function
• Determine whether an equation represents a
  function
• Evaluate a function
• Graph functions by plotting points
• Use the vertical line to identify functions

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WHAT IS A RELATION?

• A relation is a set of ordered pairs.
• Example (4,-2), (1, 2), (0, 1), (-2, 2)
• Domain is the first number in the ordered pair.
  Example(4,-2)
• Range is the second number in the ordered pair.
• Example (4,30)

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EXAMPLE 1

• Find the domain and range of the relation.
• (Smith, 1.0006), (Johnson, 0.810), (Williams,
  0.699), (Brown, 0.621)

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PRACTICE

• Find the domain and range of the following
  relation.

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HOW DO WE DETERMINE IF A RELATION IS A FUNCTION?

• A relation is a function if each domain only has
  ONE range value.
• There are two ways to visually demonstrate if a
  relation is a function.
• Mapping
• Vertical Line Test

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DETERMINE WHETHER THE RELATION IS A FUNCTION

• (1, 6), (2, 6), (3, 8), (4, 9)
• (6, 1), (6, 2), (8, 3), (9, 4)

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FUNCTIONS AS EQUATIONS

• Functions are usually given in terms of equations
  instead of ordered pairs.
• Example y 0.13x2 -0.21x 8.7
• The variable x is known the independent variable
  and y is the dependent variable.

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HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A
FUNCTION?

• x2 y 4

• Steps
• Solve the equation for y in terms of x.
• Note
• If two or more y values are found then the
  equation is not a function.

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HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A
FUNCTION

• x2 y2 4

• Steps
• Solve the equation for y in terms of x.
• Note
• If two or more y values are found then the
  equation is not a function.

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PRACTICE

• Solve each equation for y and then determine
  whether the equation defines y as a function of
  x.
• 2x y 6
• x2 y2 1

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WHAT IS FUNCTION NOTATION?

• We use the special notation f(x) which reads as f
  of x and represents the function at the number x.
• Example f (x) 0.13x2 -0.21x 8.7
• If we are interested in finding f (30), we
  substitute in 30 for x to find the function at
  30.
• f (30) 0.13(30)2 -0.21 (30) 8.7
• Now lets try to evaluate using our calculators.

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HOW DO WE EVALUATE A FUNCTION?

• F (x) x2 3x 5
• Evaluate each of the following
• f (2)
• f (x 3)
• f (-x)

• Substitute the 2 for x and evaluate.
• Then repeat.

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GRAPHS OF FUNCTIONS

• The graph of a function is the graph of the
  ordered pairs.
• Lets graph
• f (x) 2x
• g (x) 2x 4

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USING THE VERTICAL LINE TEST

• The Vertical Line Test for Functions
• If any vertical line intersects a graph in more
  than one point, the graph does not define y as a
  function of x.

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Practice

• Use the vertical line test to identify graphs in
  which y is a function of x.

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SUMMARY ANSWER IN COMPLETE SENTENCES.

• What is a relation?
• What is a function?
• How can you determine if a relation is a
  function?
• How can you determine if an equation in x and y
  defines y as a function of x? Give an example.

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AIM 1.2B WHAT KIND OF INFORMATION CAN WE OBTAIN
FROM GRAPHS OF FUNCTIONS?

• Note the closed dot indicates the graph does not
  extend from this point and its part of the graph.
• Open dot indicates that the point does not extend
  and the point is not part of the graph.

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HOW DO WE IDENTIFY DOMAIN AND RANGE FROM A
FUNCTIONS GRAPH?

• Domain set of inputs
• Found on x axis
• Range set of outputs
• Found on y -axis

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• Using set builder notation it would look like
  this for the domain
• Using Interval Notation -4, 2
• What would it look like for the range using both
  set builder and interval notation?

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IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM
ITS GRAPH

• Use Set Builder Notation.
• Domain
• Range

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IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM
ITS GRAPH
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IDENTIFYING INTERCEPTS FROM A FUNCTIONS GRAPH

• We can say that -2, 3, and 5 are the zeros of the
  function. The zeros of the function are the
• x- values that make
• f (x) 0.
• Therefore, the real zeros are the x-intercepts.
• A function can have more than one x-intercept,
  but at most one y-intercept.

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SUMMARY ANSWER IN COMPLETE SENTENCES.

• Explain how the vertical line test is used to
  determine whether a graph is a function.
• Explain how to determine the domain and range of
  a function from its graph.
• Does it make sense? Explain your reasoning.
• I graphed a function showing how paid vacation
  days depend on the number of years a person works
  for a company. The domain was the number of paid
  vacation days.

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AIM 1.3 HOW DO WE IDENTIFY INTERVALS ON WHICH A
FUNCTION IS INCREASING OR DECREASING?

• Increasing, Decreasing and Constant Functions
• A function is increasing on a open interval, I if
  
• f (x1) lt f(x2) whenever x1ltx2 for any x1 and
  x2 in the interval.

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• A function is decreasing on an open interval, I,
  if f(x1) gt f (x2) whenever x1 gt x2
• for any x1 and x2 in the interval.

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• A function is constant on an open interval, I,
  f(x1) f (x2) for any x1 and x2 in the interval.

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• Note
• The open intervals describing where function
  increase, decrease or are constant use
• x-coordinates and not y-coordinates.

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Example 1 Increases, Decreases or Constant

• State the interval where the function is
  increasing, decreasing or constant.

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Practice

• State the interval where the function is
  increasing, decreasing or constant.

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WHAT IS A RELATIVE MAXIMA?

• Definition of a Relative Maximum
• A function value f (a) is a relative maximum of f
  if there exists an open interval containing a
  such that f (a) gt f (x) for all x ? a in the open
  interval.

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WHAT IS A RELATIVE MINIMA?

• Definition of a Relative Minimum
• A function value f (b) is a relative minimum of f
  if there exists an open interval containing b
  such that f (b) lt f (x) for all x ? b in the open
  interval.

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HOW DO WE IDENTIFY EVEN AND ODD FUNCTIONS AND
SYMMETRY?

• Definition of Even and Odd Functions
• The function f is an even function if
• f (-x) f (x) all x in the domain of f.
• The right side of the equation of an even
  function does not change if x is replaced with
  x.
• The function f is an odd function if f (-x) -f
  (x)
• for all x in the domain of f.
• Every term on the right side of the equation of
  an odd function changes its sign if x is replaced
  
• with x.

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DETERMINE IF FUNCTION IS EVEN,ODD OR NEITHER

• f (x) x3 - 6x

• Steps
• Replace x with x and simplify.
• If the right side of the equation stays the same
  it is an even function.
• If every term on the right side changes sign,
  then the function is odd.

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DETERMINE IF FUNCTION IS EVEN, ODD OR NEITHER

• g (x) x4 - 2x2
• h(x) x2 2 x 1

• Steps
• Replace x with x and simplify.
• If the right side of the equation stays the same
  it is an even function.
• If every term on the right side changes sign,
  then the function is odd.

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PRACTICE

• Determine if function is Even, Odd or Neither
• f (x) x2 6
• g(x) 7x3 x
• h (x) x5 1

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• The function on the left is even.
• What does that mean in terms of the graph of the
  function?
• The graph is symmetric with respect to the
  y-axis. For every point (x, y) on the graph, the
  point (-x, y) is also on the graph.
• All even functions have graphs with this kind of
  symmetry.

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• The graph of function f (x) x3 is odd.
• It may not be symmetrical with respect to the
  y-axis. It does have symmetry in another way.
• Can you identify how?

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• For each point (x, y) there is a point (-x, -y)
  is also on the graph.
• Ex. (2, 8) and (-2, -8) are on the graph.
• The graph is symmetrical with respect to the
  origin.
• All ODD functions have graphs with origin
  symmetry.

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SUMMARYANSWER IN COMPLETE SENTENCES.

• What does it mean if a function f is increasing
  on an interval?
• If you are given a functions equation, how do
  you determine if the function is even, odd or
  neither?
• Determine whether each function is even, odd or
  neither.
• a. f (x) x2- x4 b. f (x) x(1- x2)1/2

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AIM 1.3B HOW DO WE UNDERSTAND AND USE PIECEWISE
FUNCTIONS?

• A piecewise function is a function that is
  defined by two (or more) equations over a
  specified domain.

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Example ( DO NOT COPY) READ

• A cellular phone company offers the following
  plan
• 20 per month buys 60 minutes
• Additional time costs 0.40 per minute

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HOW DO WE EVALUATE A PIECEWISE FUNCTION?
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PRACTICE

• Find and interpret each of the following
• a. C (40) b. C (80)

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HOW DO WE GRAPH A PIECEWISE FUNCTION?
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• We can use the graph of a piecewise function to
  find the range of f.
• What would the range be for the piecewise
  function? ( For previous piecewise function)

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• Some piecewise functions are called step
  functions because the graphs form discontinuous
  steps.
• One such function is called the greatest integer
  function, symbolized by int (x) or
• int (x) greatest integer that is less than or
  equal to x.
• For example
• a. int (1) 1, int (1.3) 1, int (1.5) 1,
  int (1.9) 1
• b. int (2) 2, , int (2.3) 2 , int (2.5)
  2, int (2.9) 2

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Graph of a Step Function
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FUNCTION AND DIFFERENCE QUOTIENTS

• Definition of the Difference Quotient of a
  Function
• The expression for
  h?0 is called
• the difference quotient of the function f.

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HOW DO WE EVALUATE AND SIMPLIFY A DIFFERENCE
QUOTIENT?

• If f (x) 2x2 x 3, find and simplify each
  expression
• f ( x h)
• Try

• Steps
• Replace x with (x h) each time x appears in the
  equation.

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PRACTICE

• If f (x) -2x2 x 5, find and simplify each
  expression
• f (x h)

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SUMMARY ANSWER IN COMPLETE SENTENCES.

• What is a piecewise function?
• Explain how to find the difference quotient of a
  function f,
  
• If an equation for f is given.

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AIM 1.4 HOW DO WE IDENTIFY A LINEAR FUNCTION AND
ITS SLOPE?
Data presented in a visual form as a set of
points is called a scatter plot. A scatter plot
shows the relationship between two types of data.
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• Regression line is the line that passes through
  or near the points. This is the line that best
  fits the data points.
• We can write an equation that models the data and
  allows us to make predictions.

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WHAT IS THE DEFINITION OF A SLOPE OF A LINE?
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PRACTICE

• Find the slope of a line.
• (-3, 4) and (-4, -2)
• (4, -2) and (-1, 5)

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POINT-SLOPE FORM EQUATION

• The point-slope of the equation of a nonvertical
  line with slope m that passes through the point
  (x1,y1) is
• y y1m (x x1)

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HOW DO WE WRITE AN EQUATION IN POINT-SLOPE FORM?

• Write an equation in point-slope form for the
  line with slope 4 that passes through the point
  (-1, 3).
• Then solve the equation for y.
• Steps
• Write the point-slope form equation.
• y y1m (x x1)
• Substitute the given values.
• Then solve for y.

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PRACTICE

• Write an equation in point-slope form for the
  line with slope 6 that passes through the point
  (2, -5).
• Then solve the equation for y.

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HOW DO WE WRITE AN EQUATION FOR A LINE WHEN WE
ONLY HAVE TWO POINTS?

• Write an equation in point-slope form for the
  line that passes through the points (4, -3) and
  (-2, 6).
• Then solve the equation for y.
• What would you need to do first?
• Steps
• Find the slope.
• Then choose any pair of points and substitute
  into the point-slope form equation.
• Then solve the equation for y.

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PRACTICE

• Write an equation in point-slope form for the
  line that passes through the points (-2,-1) and
  (-1, -6).
• Then solve the equation for y.

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SUMMARY ANSWER IN COMPLETE SENTENCES.

• What is the slope of a line and how is it found?
• Describe how to write an equation of a line if
  you know at least two points on that line.
• How do you derive the slope-intercept equation
  from y y1m (x x1)?

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AIM 1.4B HOW DO WE WRITE AND GRAPH LINEAR
EQUATIONS IN THE FORM OF Y MXB?

• Slope intercept equation is y mx b where m
  is the slope and b is the y-intercept of the
  equation.
• Graphing y mx b using the slope and
  y-intercept.
• Graph the y-intercept first. (0, b)
• Then use the slope to get to the other points on
  the line.
• Lets try

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EQUATIONS OF HORIZONTAL LINES

• A horizontal line has a m0. Therefore the
  equation is y0x b which can be simplified to
• y b.
• Graph y 3 or f (x) 3
• Note This is a constant function

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EQUATIONS OF VERTICAL LINES

• The slope of a vertical line is undefined. The
  equation of a vertical line is x a, where a is
  the x-intercept of the line.
• Note
• No vertical line represents a function.

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WHAT IS THE GENERAL FORM OF THE EQUATION OF A
LINE?

• Every line has an equation that can be written in
  the general form
• Ax By C or Ax By- C 0
• Where A, B, C are real numbers and A and B are
  not both zeros.

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FINDING THE SLOPE AND THE Y-INTERCEPT

• Find the slope and the y-intercept of the line
  whose equation is 3x 2y 4 0.
• Steps
• The equation is given in general form. Express in
  y mx b by solving for y.
• Then the slope and y-intercept can be identified.

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PRACTICE

• Find the slope and the y-intercept of the line
  whose equation is 3x 6y 12 0.
• Then use slope and y-intercept to graph the line.

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HOW DO WE FIND THE INTERCEPTS FROM THE GENERAL
FORM OF THE EQUATION OF A LINE?

• Graph using the intercepts 4x 3y 60
• Steps
• To find the x-intercept. Set y 0 and solve for
  x.
• To find the y-intercept. Set x 0 and solve for
  y.
• Graph the points and draw a line connecting these
  points.

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PRACTICE

• Graph using the intercepts 3x 2y 60

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REVIEW OF THE VARIOUS EQUATIONS OF LINES
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SUMMARYANSWER IN COMPLETE SENTENCES.

• How would you graph the equation x 2. Can this
  equation be expressed in slope-intercept form?
  Explain.
• Explain how to use the general form of a lines
  equation to find the lines slope and
  y-intercept.
• How do you use the intercepts to graph the
  general form of a lines equation?

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Aim 1.5 How do we find the average rate of
change?

• Slope as a rate of change
• Example 1
• The line on the graph for the number of women
  and men living alone are shown in the graph.
• Describe what the slope represents.

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• Solution Note x represents the year and y-
  number of women.
• Choose two points from the womens graph.
• Find the slope and then describe the slope.
• Remember to include the units.

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Practice

• Use the ordered pairs and find the rate of change
  for the green line or men graph. Express slope
  two decimal places and describe what it
  represents.

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Average Rate of Change

• If the graph of the function is not a straight
  line, the average rate of change between any two
  points is the slope of the line containing the
  two points.
• This line is called the secant line.

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Problem

• Looking at the graph, what is the mans average
  growth rate between the ages 13 and 18.

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The Average Rate of Change of a Function
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Example 1

• Find the average rate of change of f (x) x2
  from

• Steps
• Use

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• Find the average rate of change of f (x) x2
  from

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Average Velocity of an Object

• Suppose that a function expresses an objects
  position, s (t), in terms of time, t. The average
  velocity of the object from t1 to t2 is

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Example 2

• The distance, s (t), in feet, traveled by a ball
  rolling down a ramp is given by the function
• s (t) 5t2,
• where t is the time, in seconds after the ball
  is released. Find the balls average velocity
  from
• t1 2 seconds to t2 3 seconds
• t1 2 seconds to t2 2.5 seconds
• t1 2 seconds to t2 2.01 seconds

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Summary Answer in complete sentences.

• If two lines are parallel, describe the
  relationship between their slopes and
  y-intercept.
• If two line are perpendicular, describe the
  relationship between their slopes.
• What is the secant line?
• What is the average rate of change of a function?

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Aim 1.6 How do we recognize transformations?

• Review Algebras Common Graphs (distribute)
• Vertical Shift

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Vertical Shifts

• In general if c is positive, y f (x) c shifts
  upward c units. If c is negative it shifts
  downward c units.

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Example 1
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Practice

• Use the graph of to obtain the
  graph of

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Horizontal Shifts

• In general, if c is positive, y f (x c)
  shifts the graph of f to the left c units and y
  f (x c) shifts the graphs of f to the right c
  units.
• These are called horizontal shifts of the graph
  of f.

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Example 2
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Note
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Example 3

• Use the graph of f (x ) x2 to obtain the graph
  of h (x) (x 1)2 3.
• Steps to combining a shift
• Graph the original function.
• Then shift horizontally.
• Then shift vertically.

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Reflections of Graphs

• Reflection about the x-axis
• The graph of y - f (x) is the graph of y f
  (x) reflected about the x- axis.

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Example 4

• Use the graph of to obtain the
  graph of

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Reflections of Graphs

• Reflection about the y-axis
• The graph of y f (-x) is the graph of y f (x)
  reflected over the y axis.
• Example The point (2, 3) reflected over the
  y-axis is (-2, 3).

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Vertical Stretching

• Let f be a function and c be a positive real
  number.
• If c gt 1 the graph of y c f (x) is the graph y
  f (x) vertically stretched.
• How?
• By multiplying the y-coordinates by c.

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Vertical Shrinking

• Let f be a function and c be a positive real
  number.
• If 0 lt c lt 1 the graph of y c f (x) is the
  graph y f (x) vertically shrunk.
• How?
• By multiplying the y-coordinates by c.

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Example 6

• Use the graph of f (x) x3 to obtain the graph
  of
• h (x)

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Horizontal Shrinking

• Let f be a function and c be a positive real
  number.
• If c gt 1 the graph of y f (cx) is the graph
• y f (x) horizontal shrink.
• How?
• By dividing x- coordinates by c.

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Horizontal Stretching

• Let f be a function and c be a positive real
  number.
• If 0 lt c lt 1 the graph of y f (cx) is the graph
  
• y f (x) horizontal stretch.
• How?
• By dividing x- coordinates by c.

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Example 7

• Use the graph y f (x) to obtain each of the
  following graphs.
• a. g (x) f (2x) b. h( x) f( 1/2x)

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Sequences of Transformations

• Transformations involving more than one
  transformation can be graphed performing the
  transformations in the following order
• Horizontal shifting
• Stretching or shrinking
• Reflecting
• Vertical shifting

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Summary Answer in complete sentences.

• What must be done to a functions equation so
  that its graph is shifted vertically upward?
• What must be done to a functions equation so
  that its graph is shifted horizontally to the
  right?
• What must be done to a functions equation so
  that its graph is reflected about the x-axis?
• What must be done to a functions equation so
  that its graph is stretched vertically?

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Aim 1.7 What are composite functions?

• Finding the domain of a function (w/ no graph).
• Example 1 Find the domain of each function.

105
The Algebra of Functions

• We can combine functions by addition,
  subtraction, multiplication and division by
  performing operations with the algebraic
  expressions that appear on the right side of the
  equations.
• Example 2 Let f (x)2x 1 and g (x) x2 x
  2
• Find the following function.
• Sum (f g ) (x) f (x ) g (x)
• ( 2x 1) (x2 x 2) 1. Substitute
  given functions.
• 2. Simplify.

106
The Algebra of Functions
107
Example 3 Adding Functions and Determining the
Domain

• Find each of the following
• (f g) (x)
• The domain of (f g)

108
Composite Functions

• This is another way to combine functions.
• Example f (g (x)) can be read as f of g of x
  and it is written as

109
Example 4 Forming Composite Functions

• Given f (x) 3x 4 and g (x) x2 2x 6, find
  each of the following

110
Example 5 Forming a Composite Function
and Finding Its Domain
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Practice

• Given and g(x)
• Find each of the following
• 1.
• 2. the domain of

112
Summary Answer in complete sentences.

• Explain how to find the domain of a function of a
  radical and a rational equation.
• If equations for f and g are given, explain how
  find f- g.
• If equations for two functions are given, explain
  how to obtain the quotient of the two functions
  and its domain. ( ex. f/g(x))
• Describe a procedure for finding .
• What is the name of this function?

113
Aim 1. 8 What is an inverse function?
114
Example 1 Verifying Inverse Functions

• Show that each function is the inverse of the
  other
• Steps
• To show that f and g are inverses of each other,
  we must show that f (g (x)) x and f (g (x))
  x.
• Begin with f (g (x))
• Then with g (f (x))
• Do you get x?

115
Practice

• Show that each function is the inverse of the
  other

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How do we find the inverse?

• Find the inverse of
• f (x) 7x 5

• Steps
• Replace f (x) with y.
• Switch x and y.
• Solve for y.
• Then replace y with f-1 (x)

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How do we find the inverse?

• Find the inverse of
• f (x ) x3 1

• Steps
• Replace f (x) with y.
• Switch x and y.
• Solve for y.
• Then replace y with f-1 (x)

118
How do we find the inverse?

• Find the inverse of

• Steps
• Replace f (x) with y.
• Switch x and y.
• Solve for y.
• Then replace y with f-1 (x)

119
Practice

• Find the inverse for the following

• Steps
• Replace f (x) with y.
• Switch x and y.
• Solve for y.
• Then replace y with f-1 (x)

120
Properties of Inverse Functions

• 1. A function f has an inverse that is a
  function, f-1, if there is no horizontal line
  that intersects the graph of the function f at
  more than one point.

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Properties of Inverse Functions

• 2. The graph of a functions inverse is a
  reflection of the graph of f about the line y x.

122
Summary Answer in complete sentences.

• Explain how to determine if two functions are
  inverse of each other.
• Describe how to find the inverse of a function.
• What are some properties of a function and its
  inverse? Draw an illustration to support your
  written statement.

123
Aim 1.9 How do we find the distance and
midpoint of a segment?

• Formulas

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