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# Afm chapter 4 powerpoint

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## Afm chapter 4 powerpointPresentation Transcript

• AFM – CHAPTER 4 Functions
• 5 Function Families
• What you need to know:
• Name
• Equation
• Domain
• Range
• Linear
• Name – Constant
• Equation –
• Domain – (-  ,  )
• Range – [b]
• Linear
• Name – Oblique Linear
• Equation –
• Domain – (-  ,  )
• Range – (-  ,  )
• Power Functions
• Equation –
• Domain – (-  ,  )
• Range – [0,  )
• Power Functions
• Name – Cubic
• Equation –
• Domain – (-  ,  )
• Range – (-  ,  )
• Root Functions
• Name – Square root
• Equation –
• Domain – [0,  )
• Range – [0,  )
• Root Functions
• Name – Cube root
• Equation –
• Domain – (-  ,  )
• Range – (-  ,  )
• Reciprocal Functions
• Name – Rational
• Equation –
• Domain –(-  ,0)  (0,  )
• Range – (-  ,0)  (0,  )
• Reciprocal Functions
• Name – Rational Squared
• Equation –
• Domain – (-  ,0)  (0,  )
• Range – (0,  )
• Absolute Value Function
• Name – Absolute value
• Equation –
• Domain – (-  ,  )
• Range – [0,  )
• Greatest Integer Function
• Name – Greatest Integer
• Equation –
• Domain – (-  ,  )
• Range – (integers)
• Trig Functions
• Name – Sine
• Equation –
• Domain – (-  ,  )
• Range – [-1,1]
• Trig Functions
• Name – Cosine
• Equation –
• Domain – (-  ,  )
• Range – [-1,1]
• Vertical Line Test
• A curve in the coordinate plane is the graph of a function iff no vertical line intersects the curve more than once.
• Piecewise Functions
• Sketch the graph of:
• Equations That Define Functions
• Does the equation define y as a function of x ?
• Homework 4.2
• Page 228 – 230; 2 – 50 even 60 –70 even
• Know Family of Functions graphs
• Name, equation, domain, range
• 4.4 – Average Rate of Change
• Average rate of change of a function f(x) between x = a and x = b is:
• Slope of secant line drawn between x=a & x=b or line through the points (a,f(a)),(b,f(b)).
• Finding the Average Rate of Change
• Find the average rate of change if
• Example 2
• An object is dropped from a height of 3000 feet, its distance above the ground h , after t seconds is given by:
• Find the average speed
• Between 1 & 2 seconds
• Between 4 & 5 seconds
• Increasing and Decreasing Functions
• A function f is increasing if:
• A function f is decreasing if:
• State the intervals on which the function whose graph is shown is increasing or decreasing.
• Sketch the graph of the function:
• Find the domain and range of the function
• Find the intervals on which f increases and decreases
• Determine the average rate of change of the function between the given values of the variable.
• Use a graphing calculator to draw the graph of f . State approximately the intervals on which f is increasing and on which f is decreasing.
• The table gives the population in a small coastal community for the period 1990 – 1999. Figures shown are for January 1 in each year.
• What was the average rate of change of population between 1991 and 1994?
• What was the average rate of change of population between 1995 and 1997?
• For what period of time was the population increasing?
• For what period of time was the population decreasing?
• Transformations
• Vertical Shift
• Horizontal Shift
• Reflecting
• Stretching/Shrinking
• Exploring transformations
• Graph
• Graph
• More transformations
• Graph:
• Graph:
• General Rules for Transformations
• Vertical shift:
• y=f(x) + c  c units up
• y=f(x) – c  c units down
• Horizontal shift:
• y=f(x+c)  c units left
• y=f(x-c)  c units right
• Reflection:
• y= – f(x)  reflect over x -axis
• y= f(-x)  reflect over y -axis
• Stretch/Shrink:
• y=af(x)  (a > 1) Stretch vertically
• y=af(x)  (0 < a < 1) Shrink vertically
• Even & Odd Functions
• Algebraically:
• Even – f is even if f(-x) = f(x)
• Odd – f is odd if f(-x) = - f(x)
• Graphically:
• Even – f is even if its graph is symmetric to the y -axis
• Odd – f is odd if its graph is symmetric to the origin
• Determine Algebraically if the function is even, odd or neither
• Use the rules of transformations to graph the following:
• 4.7 Combining Functions
• Combining – Addition, Subtraction, Multiplication, or Division
• Composition of functions – Putting two functions together using the rules of one of the functions
• Combining Functions
• Addition/Subtraction – f(x) and g(x)
• (f ± g)(x) = f(x) ± g(x)  Add/Subtract, then combine like terms
• Domain: D:f(x)  D:g(x)
• Multiplication – f(x) and g(x)
• (fg)(x) = f(x) ·g(x)  Multiply, then combine like terms
• Domain: D:f(x)  D:g(x)
• Division – f(x) and g(x)
•  Divide, then simplify
• Domain: D:f(x)  D:g(x), where g(x)  0
• Examples Let Find Domain f(x)  Domain g(x)  Find
• Composition of Functions
• Examples If : and Find:
• Composition of 3 Functions Find: If: Page 276 - # 23,25,27
• Variation
• Direct Variation
• Indirect Variation
• Joint Variation
• Direct Variation
• y varies directly as x
• y is directly proportional to x
• y is proportional to x
• Formula (Equation to use)
• y = kx 
;k is constant of proportionality
• During a thunderstorm, the distance between you and the storm varies directly as the time interval between the lightening and thunder.
• Suppose thunder from a storm 5400 ft away takes 5 seconds reach you.
• Determine the constant of proportionality and write the variation equation for the model.
• Sketch the graph. What does the k represent?
• If the time interval between the lightening and thunder is 8 sec. How far away is the storm?
• Inverse Variation
• y varies inversely as x
• y is inversely proportional to x
• Formula (Equation to use)
• Boyle’s Law – When a sample of gas is compressed at a k onstant temperature the pressure of the gas is inversely proportional to the volume of the gas.
• P – pressure
• v – volume
• k – constant of proportionality
• Suppose the pressure of a sample of air that occupies 0.106 m ³ @ 25ºC is 50 kPa. Find the constant of proportionality and write the equation that expresses the inverse proportionality
• If the sample expands to a volume of .3m³, find the new pressure
• Joint Variation
• Used when a quantity depends on more than one other quantity.  It depends on them jointly.
• z varies jointly as x and y
• z is jointly proportional to x and y
• z is proportional to x and inversely proportional to y
• Newton’s Law of Gravitation – Two objects with masses m 1 and m 2 attract each other with a force F that is jointly proportional to their masses and inversely proportional to the square of the distance r between the objects. Express Newton’s Law of Gravitation as an equation.
• Examples of Variation
• Write an equation that expresses the statement:
• R varies directly as t .
• v is inversely proportional to z .
• y is proportional to s and inversely proportional to t .
• R is proportional to j and inversely proportional to the squares of s and t .
• Express the statement as a formula. Use the given information to find the constant of proportionality
• y is directly proportional to x . If x = 4, then y = 72.
• M varies directly as x and inversely as y . If x = 2 and y = 6, then M = 5.
• s is inversely proportional to the square root of t . If s = 100, then t = 25.
• Hooke’s Law states that the force F needed to keep a spring stretched x units beyond its natural length is directly proportional to x . Here the constant of proportionality is called the spring constant.
• Write Hooke’s Law as an equation.
• If a spring has a natural length of 10 cm and a force of 40 N is required to maintain the spring stretched to a length of 15 cm, find the spring constant.
• What force is needed to keep the spring stretched to a length of 14 cm?
• The resistance R of wire varies directly as its length L and inversely as the square of its diameter d .
• Write an equation that expresses this joint variation.
• Find the constant of proportionality if a wire 1.2 m long and 0.005 m in diameter has a resistance of 140 ohms.
• Find the resistance of a wire made of the same material that is 3 m long and has a diameter of 0.008 m.
• Modeling Quadratic & Cubic Functions
• Define the variable
• Find the equation (model)