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JEOPARDY!Definiti
ons
Rational and Equivalent Functions
Functions Transformations
Inverse Functions
10 10 10 10 10
20 20 20 20 20
30 30 30 30 30
40 40 40 40 40
50 50 50 50 50
Definitions – 10Answer
The domain of a function is the set of all x values that produce a y value.
The range of a functions is set of all y values that result from any function.
Definitions – 30Answer
• Horizontal reflection and stretch• Horizontal shift• Vertical reflection and stretch• Vertical shift
Definitions – 50Answer
k – reflection if k is negativek – horizontal stretch by a factor of 1/kp – horizontal shifta – reflection if a is negativea – vertical stretch by a factor of aq – vertical shift
Rational and Equivalent Functions - 10
Are the two expressions equivalent?
(x-3)2(x+4)(x+1)
x4- x3 -17x2 +21x+36
Rational and Equivalent Functions – 10Answer
Yes
Solution: Test at least 3 points in both functions to make sure they are they return the same value.
For example, test x=0
Function 1: (x-3)2(x+4)(x+1)=(0-3) 2(0+4)(0+1)=(-3) 2(4)(1)=36
Function 2: x4- x3 -17x2 +21x+36=04-03-17(0) 2+21(0)+36=36
Rational and Equivalent Functions - 20
Simplify the function and state all restrictions.
-x2 -7x+8 . x +5 x+8 9x-9
Rational and Equivalent Functions – 20Answer
Solution: -x2 -7x+8 . x +5 x+8 9x-9= -(x-1)(x+8) . x+5 x+8 9(x-1)= -(x+5) 9
-(x+5) 9
x ≠ -8, 1
Rational and Equivalent Functions – 30Answer
Solution:y = 2(x-4)2 -7 = 2(x2 -8x+16) -7 = 2x2 -16x+32-7 = 2x2 -16x+25
y = 2x2 -16x + 25
Rational and Equivalent Functions - 40
Simplify the function and state all restrictions.
x+3 ÷ (x-1)(x+3)x+2 (x-1)2
Rational and Equivalent Functions – 40Answer
Solution:x+3 ÷ (x-1)(x+3)x+2 (x-1)2
=x+3 . (x-1)2__ x+2 (x-1)(x+3)= x-1 x+2
x - 1 x+ 2
x ≠ -2, 1, -3
Rational and Equivalent Functions - 50
Simplify the function and state all restrictions.
5 - x _ x2 -5x 5x-25
Rational and Equivalent Functions – 50Answer
Solution: 5 - x _ x2 -5x 5x-25= 5 - x x(x-5) 5(x-5)= 5 . 5- x . x x(x-5) 5 5(x-5) x= 25 - x2 5x(x-5) 5x(x-5)= 25-x2
5x(x-5)= - (x2-25) 5x(x-5)= -(x+5)(x-5) 5x(x-5)= -(x+5) 5x
Find a common denominator: 5x(x-5)
-(x+5) 5x
x ≠ 0, 5
Transformations - 10
If you start at a point (3, 5) and move 4 units left and 3units up, what is the new coordinate?
Transformations – 20Answer
• Shift 5 units right• Reflection in the x-axis (vertical reflection)• Vertical stretch by a factor of 3• Shift 1 unit up
Transformations - 30
Choose any three base points and write them after thetransformation.
f(x) = 5√-2(x+1) ______
Transformations – 30Answer
(0, 0) (-1, 0) (1, 1) (-1.5, 5) (4, 2) (-5, 10) (9, 3) (-5.5, 15) (16, 4) (-9, 20)
(x, y) (- x, y) (- x-1, y) (-2s+1, 5y)
(0, 0) (0, 0) (-1, 0) (-1, 0)(1, 1) (-0.5, 1) (-1.5, 1) (-1.5, 5)(4, 2) (-4, 2) (-5, 2) (-5, 10)(9, 3) (-4.5, 3) (-5.5, 3) (-5.5, 15)(16, 4) (-8, 4) (-9, 4) (-9, 20)
Inverse Functions – 20Answer
f -1(x)= 3x-12
Solution:y = (1/3 )x +4x = (1/3 )y+4x-4 = (1/3 )y3(x-4) =y
f -1 (x) = 3(x-4) = 3x-12
Inverse Functions – 30Answer
f -1(x)= -(1/4)(x – 3)2
Solution:f(x) = 2√-x +3y = 2√-x +3x = 2√-y+3x-3 = 2√-yx-3 = √-y 2x-3 2 = -y 2 x-3 2 = y 2
f -1(x) = x-3 2
2
( )
-( )
-( )
______
____
Inverse Functions – 40Answer
f -1(x)= 4±√(1/2)(x+3)
Solution:f(x) = 2x2+16x+29y = 2x2+16x+29y = 2(x2+8x) +29y = 2(x2 +8x +16-16) +29y = 2(x+4) 2 -32+29y = 2(x+4) 2 -3x = 2(y+4) 2 -3X+3 = 2(y+4) 2 X+3 = (y+4) 2 2X+3 = y+4 2X+3 -4 = y 2
________
±√__
__±√ f -1 (x) =-4 x+3
2±√ __
Inverse Functions - 50
Sketch the original and the inverse of the followingfunction (without finding the inverse). Is the inverse afunction?
HINT - draw the y=x line and then draw the reflection
f(x) = -(3(x-5))2 -1