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1.3 - New Functions From 1.3 - New Functions From Old FunctionsOld Functions

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Translation: f (x) + k

y2 Direction of Translation

Units Translated

Value of k

x2 – 4

x2 – 2

x2 + 2

x2 + 4

Graph y1 = x2 on your graphing calculator and then graph y2 given below to determine the movement of the graph of y2 as compared to y1. Generalize the effect of k.

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Translation: f (x – h)

y2 Direction of Translation

Units Translated

Value of h

(x – 4)2

(x – 2)2

(x + 2)2

(x + 4)2

Graph y1 = x2 on your graphing calculator and then graph y2 given below to determine the movement of the graph of y2 as compared to y1. Generalize the effect of h.

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• If h < 0, the graph shifts h units _______. If h > 0, the graph shifts h units _______. The value of h causes a _________________ translation.

• If k > 0, the graph shifts k units _______. If k < 0 then the graph shifts k units ______. The value of k causes a ______________ translation.

Horizontal and Vertical Translations f (x – h) + k

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Examples

Use the library of functions to sketch a graph of each of the following without using your graphing calculator.

(a) f(x) = (x + 3)2 – 2

(b) g(x) = | x – 2 | + 3

(c)

(d)

21)( xxh3( ) ( 3) 1h x x

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x- and y-Axis Reflections

• The graph of y = - f(x) is the same as graph of f(x) but reflected about the __-axis.

• The graph of y = f(-x) is the same as graph of f(x) but reflected about the __-axis.

xy 1

xy 2

xy 2

Graph with your graphing calculator:

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Compression and Stretches

• If |a| > 1, the graph of y = af(x) is ______________ vertically or _____________ horizontally…

• 0 < |a| < 1 the graph of y = af(x) is _____________ vertically or _____________ horizontally…

• … as compared to y = f(x).

Sketch the following using your graphing calculatory1 = |x| y2 = 3|x| y2 = (⅓) |x|

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Examples

Use the library of functions to sketch a graph of each of the following without using your graphing calculator.

(a) f(x) = - (x + 3)2 – 2

(b) g(x) = 2| x – 1 | + 3

(c)

(d)

12( ) 1 2h x x

3( ) 2( 3) 1h x x

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The Algebra of Functions

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The sum f + g is the function defined by

(f + g)(x) = f(x) + g(x)

The domain of f + g consists of numbers x that are in the domain of both f and g (the intersection of the domains).

The difference f - g is the function defined by

(f - g)(x) = f(x) - g(x)

The domain of f - g consists of numbers x that are in the domain of both f and g (the intersection of the domains).

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The product f ∙ g is the function defined by

(f ∙ g)(x) = f(x) ∙ g(x)

The domain of f ∙ g consists of numbers x that are in the domain of both f and g (the intersection of the domains).

The quotient f / g is the function

The domain of f / g consists of all x such that x is in the domain of f and g and g(x) ≠ 0 (the intersection of the domains).

( )( )

( )

f f xx

g g x

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Given two functions f and g, the composite function is defined by

(f ◦ g)(x) = f(g(x))Read “f composite g of x”

The domain of f ◦ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. Note: In general (f ◦ g)(x) ≠ (g ◦ f)(x)

Composition of Functions

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xg(x)

f(g(x))

g f

Domain of g Domain of f

Range of g

Range of f

(f ◦ g)(x) = f(g(x))Or f(g)

Domain of f(g) Range of f(g)

(equal to or a subset of)

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= 2 – 3

Composition of Functions

Let f(x) = 2x – 3 and g(x) = x 2 – 5x. Determine (f ◦ g)(x).

(f ◦ g)(x) = f(g(x)) = f(x 2 – 5x)

x(x 2 – 5x)

= 2x 2 – 10x – 3

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Examples

(a) Let and g(x) = x2 – 2. Determine (i)

(f ◦ g)(x) and (ii) (g ◦ f)(x). Determine the domains of each.

(b)If y = cos (x2 – 2) and y = (f ◦ g)(x), determine f and g.

(c) If y = esin(x+5) and y = (f ◦ g ◦ h)(x), determine f, g, and h.

1

1)(

x

xxf