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Types of Functions Constant, Linear and Non-Linear.

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Presentation on theme: "Types of Functions Constant, Linear and Non-Linear."— Presentation transcript:

1 Types of Functions Constant, Linear and Non-Linear

2 7/9/2013 Types of Functions 2 2 Constant Function Each member of a set (domain) maps to the same member of another set (range) A = domain B = range a b k { } c, … (, ) c k,, b k a k F =

3 7/9/2013 Types of Functions 3 3 Constant Function Examples f = { (-4, 3), (-1, 3), (2, 3), (4, 3), (7, 3) } x y     x y Domain = { -4, -1, 2, 4, 7 } Range = { 3 }

4 7/9/2013 Types of Functions 4 4 Constant Function Examples f = { (x, 2)  x  R } x y y = 2 y = f (x) = 2 Note: No table Why ? Domain = R Range = { 2 }

5 7/9/2013 Types of Functions 5 5 Linear Functions Always of form for constants a and b f (x) = a x + b Question: What if a = 0 ? What properties does f (x) have ?

6 7/9/2013 Types of Functions 6 6 Linear Functions Changes in f (x) = a x + b Always proportional to changes in x f (x) changes by a per unit change in x Rate of change of f (x) Rate is a, the slope of the graph f is linear iffi rate of change is constant

7 7/9/2013 Types of Functions 7 7 Linear Functions Changes in f (x) = a x + b Always proportional to changes in x f (x) changes by a per unit change in x

8 7/9/2013 Types of Functions 8 8 Linear Functions Rate of change of f (x) = a x + b Rate is a, the slope of the graph f is linear if and only if rate of change is constant

9 7/9/2013 Types of Functions 9 9 Linear Function Examples f (x) = 2x + 1, for x  A where      A = { -2, -1, 1, 3, 4 } -2 x y = f (x) x y Discrete Graph Question: Is rate of change constant ?

10 7/9/2013 Types of Functions 10 Types of Functions Linear Function Examples f (x) = 2x + 1, for all x  R Domain = R Range = R x y = f (x) y = 2x + 1 Question: Is rate of change constant ? Continuous Graph

11 7/9/2013 Types of Functions 11 Rate of Change and Slope Rate of Change Example f (x) = 2x + 1 for x  R How does f (x) change as x changes ? Rate of change of f (x) relative to x is x y = f (x) y = 2x + 1   x1x1 x2x2   y1y1 y2y2 ∆x ∆y ∆f∆f ∆x ∆y ∆x =  

12 7/9/2013 Types of Functions 12 Rate of Change and Slope Rate of Change Example x y = f (x) y = 2x + 1   x1x1 x2x2   y1y1 y2y2 ∆x ∆y y 2 – y 1 x 2 – x 1 = ∆f∆f ∆x ∆y ∆x =   (2x 2 + 1) – (2x 1 + 1) x 2 – x 1 = 2x 2 – 2x – 1 x 2 – x 1 = = 2 f (x) = 2x + 1 f (x) = 2x + 1 for x  R

13 7/9/2013 Types of Functions 13 Rate of Change We now have ∆ f = ∆y = 2 ∆x Change in f (x) is twice the change in x … Rate of Change and Slope ∆f∆f ∆x ∆y ∆x = = 2 x y = f (x) y = 2x + 1   x1x1 x2x2   y1y1 y2y2 ∆x ∆y   f (x) = 2x + 1 for x  R … at any x

14 7/9/2013 Types of Functions 14 Intercepts from Formulas Suppose we have a line L with non-zero slope m and slope-intercept formula y = mx + b Linear Functions in General x y   Vertical Intercept (0, b) Slope Horizontal Intercept L ( ) b m – 0, b m – 0,

15 7/9/2013 Types of Functions 15 x y Find intercepts and graph for y = x + 2  Note: y = mx + b m = 1 and b = 2  y-intercept : (0,b) = (0,2) Linear Function Example  (0, 2)

16 7/9/2013 Types of Functions 16  y-intercept : (0,b) = (0,2) Slope : m = 1  x-intercept : x y Linear Function Example  (0, 2)  (–2, 0) ( ), b m – 0 = 2 1 –, 0 = – 2 0, m = 1 and b = 2

17 7/9/2013 Types of Functions 17 x y Find the equation  Intercepts (0,b) = (0, 3) Linear Functions Example  = ( ) 4, 04, 0 (4, 0)  b = 3 b m – = 4 and General equation y = mx + b y = x – becomes (0, 3) ( ) b m –, 0 = m b 4 – 3 4 – =

18 7/9/2013 Types of Functions 18 Average Rate of Change For function f (x) average rate of change of f (x) over any interval [ x 1, x 2 ] is f (x 2 ) – f (x 1 ) x 2 – x 1 = ∆f∆f ∆x y x  y = x  x1x1 x2x2   ∆f∆f ∆x NOTE: Rate not constant … depends x 1 and x 2 So f not linear !

19 7/9/2013 Types of Functions 19 Average Rate of Change Example: f (x) = x y x  y = x  x1x1 x2x2 ∆f∆f ∆x ∆f∆f (x ) – (x ) x 2 – x 1 = x 2 2 – x 1 2 x 2 – x 1 = (x 2 + x 1 )(x 2 – x 1 ) x 2 – x 1 = Rate of change, i.e. slope, is not constant   NOTE: x 1 + x 2 = ∆f∆f ∆x … depends on x 1 and x 2

20 7/9/2013 Types of Functions 20 Average Rate of Change Example: f (x) = x y x  y = x  x1x1 x2x2 ∆f∆f ∆x   ∆x = 5 5 ∆f∆f = x 1 + x 2 = ∆f∆f ∆x = ∆f∆f 5 5 =1 On interval [-2, 3] and ∆x = 5 25 ∆f∆f = = ∆f∆f ∆x 5 25 =5 On interval [0, 5] and Rate of change, i.e. slope, is not constant … so f is not linear NOTE:

21 7/9/2013 Types of Functions 21 Any function f (x) NOT of form Example 1: f (x) = x Nonlinear Functions f (x) = a x + b x y     Average rate of change (slope) not constant … P1P1 P2P2 P3P3 P4P4 so function not linear

22 7/9/2013 Types of Functions 22 P1P1 P2P2 Any function f (x) NOT of form Example 2: Nonlinear Functions f (x) = a x + b x y     Average rate of change (slope) not constant … P3P3 P4P4 so function not linear g (x) | x =

23 7/9/2013 Types of Functions 23 Any function f (x) NOT of form Example 3: Nonlinear Functions f (x) = a x + b x y Average rate of change (slope) not constant … P1P1 P3P3 P4P4 so function not linear h(x) = 1 1 – x, x < 1     P2P2 Asymptote

24 7/9/2013 Types of Functions 24 The Difference Quotient Consider a nonlinear function f Secant line through (x, f (x)) and (x + h, f (x + h)) has slope m y = f (x) x  x (x, f (x)) h x + h (x + h, f (x + h)) Slope m m = f (x + h) – f (x) (x + h) – x = f (x + h) – f (x) h = ff xx ff =  x   xx

25 7/9/2013 Types of Functions 25 The Difference Quotient Consider a nonlinear function f Secant line through (x, f (x)) and (x + h, f (x + h)) has slope m y = f (x) x  x (x, f (x)) h x + h (x + h, f (x + h)) Slope m m = f (x + h) – f (x) h ff =  x   xx Difference quotient for f on [ x, x+h ], i.e. the average rate of change of f from x to x + h

26 7/9/2013 Types of Functions 26  More Secants  What happens to m as h gets smaller and smaller? What is the slope of the tangent line at x ? What do the secants approach as The Difference Quotient y = f (x) x  x x+hx+h  m = f (x + h) – f (x) h  x+hx+h   x+hx+h  As h 0 m ?, Slope m Tangent Line h 0 ? 

27 7/9/2013 Types of Functions 27 The Difference Quotient Example: f (x) = x 2 – 4 x y = ((x + h) 2 – 4) – (x 2 – 4) (x + h) – x = x 2 + 2xh + h 2 – 4 – x h = 2xh + h 2 h = h(2x + h) h f (x + h) – f (x) (x + h) – x m = = 2x + h m Thus m

28 7/9/2013 Types of Functions 28 The Difference Quotient Example: f (x) = x 2 – 4 x y   m 2x as h 0 Clearly, Slope m depends only on the value of x chosen At x = 1, 2x = 2 At x = 3, 2x = 6 2x + h = m Conclusion ? (1, –3) (3, 5)

29 7/9/2013 Types of Functions 29 Think about it !

30 7/9/2013 Types of Functions 30 x y Find intercepts and graph for y = x + 2  Note: y = mx + b m = 1 and b = 2  Slope : 1  Intercepts : (0,b) = (0,2) Linear Function Example  (0, 2)  (–2, 0) ( ), b m – 0 = 2 1 –, 0 = – 2 0,

31 7/9/2013 Types of Functions 31  More Secants  What happens to m as h gets smaller and smaller? What is the slope of the tangent line at x ? The Difference Quotient y = f (x) x  x x+hx+h  m = f (x + h) – f (x) h  x+hx+h   x+hx+h  As h 0 m ?, Slope m 

32 7/9/2013 Types of Functions 32  More Secants  What happens to m as h gets smaller and smaller? What is the slope of the tangent line at x ? What do the secants approach as The Difference Quotient y = f (x) x  x m = f (x + h) – f (x) h As h 0 m ?,  Tangent Line h 0 ?

33 7/9/2013 Types of Functions 33 Types of Functions Linear Function Examples f (x) = 2x + 1, for x  A where      A = { -2, -1, 1, 3, 4 } x y = f (x) x y Discrete Graph Question: Is rate of change constant ?

34 7/9/2013 Types of Functions 34 Types of Functions Linear Function Examples f (x) = 2x + 1, for x  A where      A = { -2, -1, 1, 3, 4 } x y = f (x) x y Discrete Graph Question: Is rate of change constant ?


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