Function Family Album
12
Function Family Album
Meghan Nagia
Absolute Value Parent Function
Absolute Value Function
Equation: f(X) = |X|
Domain: (-∞, ∞)
Range: [0, ∞)
X-Int: (0, 0)
Y-Int: (0, 0)
End Behavior:
As X--> ∞ , f(X)--> ∞
As X--> -∞ , f(X)--> ∞
A shift to the right 3
End Behavior:
As X--> ∞ , f(X)--> ∞
As X--> -∞ , f(X) --> ∞
Equation: f(x) = |X + 3|
Domain: (- ∞ , ∞ )
Range: [0, ∞ )
X-Int: (3, 0)
Y-Int: (0, 3)
Translation Right
Translation UP
A shift Up 4
Equation: f(X) = |X| + 4
Domain: (- ∞ , ∞ )
Range: [4, ∞ )
X-Int: None
Y-Int: (0, 4)
End Behavior:
As X--> ∞ , f(X)--> ∞
As X--> -∞ , f(X)--> ∞
A Vertical Flip - Reflection over the X-Axis
Equation: f(X) = -|X|
Domain: (- ∞ , ∞ )
Range: (- ∞ , 0]
X-Int: (0, 0)
Y-Int: (0,0)
End Behavior:
As X--> ∞ , f(X)--> -∞
As X--> -∞, f(X)--> -∞
Reflection
Vertical
Stretch
Vertical Stretch By a Factor of 5
Equation: f(X) = 5|X|
Domain: (- ∞ , ∞ )
Range: [0, ∞ )
X-Int: None
Y-Int: (0, 0)
End Behavior:
As X--> ∞ , f(X)--> ∞
As X--> -∞ , f(X)--> -∞
Solving Absolute Value Equations
Steps:
1. Move Constants to 1 Side
2. Multiply/divide to get absolute value alone
3. Write 3 equations: 1 positive & 1 negative
4. Solve & Check Work
Equations & Steps
|3X - 13| + 9 = 32
|3X -13| = 23
3X - 13 = 23 3X - 13 = -23
3X = 36 3X = -10
X = 12 X = -10/3 (-3.33)
4|X - 5| +10 = 30
4|X - 5| = 20
|X - 5| = 5
X - 5 = 5 X - 5 = -5
X = 10 X = 0
Real World Example
Absolute Value in the Real World
Birds in Formation
This is a picture of birds flying in formation. Typically, most birds fly in a "V" formation, which is a good example of an absolute value function because the shape of an absolute value function is also a "V".
Exponential Parent Function
Equation: f(X) = 2^X
Domain: (-∞, ∞)
Range: (0, ∞)
X-Int: None
Y-Int: (0, 1)
End Behavior:
As X--> ∞, f(x)--> ∞
As X--> -∞, f(X)--> 0
Exponential Function
Asymptote: y= 0
Translation
left
A Shift to the left 5
Equation: f(X) = 2^(X + 5)
Domain: (-∞, ∞)
Range: (0, ∞)
X-Int: None
Y-Int: (0, 32) f(X) = 2^(0 + 5) --> 2^5=32
End Behavior:
As X--> ∞, f(X)--> ∞
As X--> -∞, f(X)--> 0
Asymptote: y= 0
A shift down 4
Equation: f(X) = 2^X - 4
Domain: (-∞, ∞)
Range: (-4, ∞)
X-Int: (2, 0)
Y-Int: (0, -3)
End Behavior:
As X--> ∞, f(X)--> ∞
As X--> -∞, f(X)--> -4
Translation Down
Asymptote: y= -4
Reflection
A Horizontal Flip
Reflection over the Y-Axis
Equation: f(X) = 2^(-X)
Domain: (-∞, ∞)
Range: (0, ∞)
X-Int: None
Y-Int: (0, 1)
End Behavior:
As X--> ∞, f(X)--> 0
As X--> -∞, f(X)--> ∞
Asymptote: y= 0
Vertical Shrink by factor of 1/4
Vertical Shrink
Equation: f(X) = (1/4) 2^X
Domain: (-∞, ∞)
Range: (0, ∞)
X-Int: None
Y-Int: (0, 1/4)
End Behavior:
As X--> ∞, f(X)--> ∞
As X--> -∞, f(X)--> 0
Asymptote: y= 0
Equation & Steps
Solving Exponential Equations
Steps:
1. Isolate the base and exponent
2. Rewrite both sides of the equation with the same base
3. Set the exponents equal to each other
4. Solve
9^(2X+ 1) = 243 ^(4X - 6)
3^2(2X + 1) = 3^5(4X - 6)
2(2X + 1) = 5(4X - 6)
4X + 2 = 20X -30
32 = 16X
X = 2
6^(3X + 7) = 1/216 ^(2X - 12)
6^(3X + 7) = 6^-3(2X - 12)
3X +7 = -3(2X - 12)
3X + 7 = 6X -36
-3X = -43
X = 43/3 (14.33)
Exponential function in real life
Real World Example
Eiffel Tower
This is the Eiffel Tower located in Paris, France. The sides of the Eiffel Tower are good examples of an exponential function. The shape of the tower sideways is an upwards curve, which is similar to the shape of an Exponential function.
Logarithmic Parent Function
2
Equation: f(X) = log X
Domain: (0, ∞)
Range: (-∞, ∞)
X-Int: (1/2, 0)
Y-Int: None
End Behavior:
As X--> ∞, f(X)--> ∞
As X--> 0, f(X)--> -∞
Translation Up
Asymptote: X= 0
Logarithmic Function
Translation Right
A Shift to the Right 5
2
Equation: f(X) = log (X - 5)
Domain: (5, ∞)
Range: (-∞, ∞)
X-Int: (11/2, 0)
Y-Int: None
End Behavior:
As X--> ∞, f(X)--> ∞
As X--> 5, f(X)--> -∞
Asymptote: X = 5
A Shift Up 3
2
Equation: f(X) = log X + 5
Domain: (0, ∞)
Range: (-∞, ∞)
X-Int: (1/8, 0)
Y-Int: None
End Behavior:
As X--> ∞, f(X)--> ∞
As X--> 0, f(X)--> -∞
Asymptote: X = 0
A Vertical Flip
Reflection over the x-axis
2
Reflection
Equation: f(X) = -log X
Domain: (0, ∞)
Range: (-∞, ∞)
X-Int: (1, 0)
Y-Int: None
End Behavior:
As X--> ∞, f(X)--> -∞
As X--> 0, f(X)--> ∞
Asymptote: X = 0
Vertical Stretch by Factor of 6
2
Vertical Stretch
Equation: f(X) = 6 log X
Domain: (0, ∞)
Range: (-∞, ∞)
X-Int: (1, 0)
Y-Int: None
End Behavior:
As X--> ∞, f(X)--> ∞
As X--> 0, f(X)--> -∞
Asymptote: X= 0
Solving Logarithmic Equations
Steps:
1. Get logs on 1 side
2. Condense the logs (using logarithm properties)
3. Exponentiate (using the base)
4. Solve
4
Equations & Steps
4
2 log X = log 32 + log 8
0 = log 32 + log 8 - 2 log X
0 = log 32 + log 8 - log X^2
0 = log (32 x 8) / X^2
0 = log (256/X^2)
4^0 = 4 log (256/X^2)
1 = 256/X^2
X^2 = 256
X = +/- 16
X = 16
3
4
3
log 8 + log (X + 2) = 4
log 8(X + 2) = 4
3 log 8(X + 2) = 3^4
8X + 16 = 81
8X = 65
X = 65/8 (8.125)
Logarithmic Function In real Life
Real World Example
Wilted Tree
This is a picture of a wilted tree. The shape of leaves drooping on a wilted tree is similar to the downward curve shape of a logarithmic function.
Real world example
Quadratic Parent Function
Quadratic function
Equation: f(X) = X^2
Domain: (-∞, ∞)
Range: [0, ∞)
X-Int: (0, 0)
Y-Int: (0, 0)
End Behavior:
As X--> ∞, f(x)--> ∞
As X--> -∞, f(X)--> ∞
Translation right
A Shift to the Right 7
Equation: f(X) = (X - 7)^2
Domain: (-∞, ∞)
Range: [0, ∞)
X-Int: (7, 0)
Y-Int: (0, 49) -->(0 - 7)^2 = -7^2
End Behavior:
As X--> ∞, f(x)--> ∞
As X--> -∞, f(X)--> ∞
A shift down 4
End Behavior:
As X--> ∞, f(x)--> ∞
As X--> -∞, f(X)--> ∞
Equation: f(X) = X^2 - 4
Domain: (-∞, ∞)
Range: [4, ∞)
X-Int: (2, 0) & (-2, 0)
Y-Int: (0, 4)
Translation Down
Reflection
A vertical flip - Reflection over the x-axis
Equation: f(X) = -X^2
Domain: (-∞, ∞)
Range: [0, -∞)
X-Int: (0, 0)
Y-Int: (0, 0)
End Behavior:
As X--> ∞, f(x)--> -∞
As X--> -∞, f(X)--> -∞
Horizontal
Stretch
Horizontal stretch by 1/4
Equation: f(X) = (1/4X)^2
Domain: (-∞, ∞)
Range: [0, ∞)
X-Int: (0, 0)
Y-Int: (0, 0)
End Behavior:
As X--> ∞, f(x)--> ∞
As X--> -∞, f(X)--> ∞
Solving quadratic equations
Steps:
1. Rewrite in standard form (Ax^2 + Bx + C)
2. Take out the GCF
3. Factor completely using factor methods (difference of squares, difference of cubes, etc)
If factoring is not possible, use the quadratic method ( -b +/- square root of b^2 - 4AC ALL over 2A)
4. Use the Zero Product Rule (set factored equation equal to zero)
5. Solve
2 + 2 7 2 - 2 7
√
__________
4
X = 1.8, X = -0.8
√
______________________
2(2)
12X^2 + 23X + 11X + 16 + 4 = 0
12X+2 + 34X + 20 = 0
2( 6X^2 + 17X + 10) = 0
(X + 2) (6X + 5) = 0
(X + 2) = 0 (6X + 5) = 0
X = -2, X = -5/6
7X^2 + 9X^2 - 11X - 5X+ 20 + 4
16X^2 - 16X +24
8( 2X^2 - 2X + 3)
2 +/- ( 2^2 - 4(-2)(3) )
2 +/- 28
√
________________
4
Quadratic function in the real world
Equations & Steps
Moon Bridge
This is a moon bridge at the Japanese Tea Gardens in San Francisco. The bridge is shaped like an upside down U, which is similar to the shape of a quadratic function, which is parabolic. More specifically, this bridge more closely resembles a quadratic function that has been reflected over the X-axis.
rational parent function
Equation: f(X) = 1/X
Domain: (-∞, 0) U (0, ∞)
Range: (-∞, 0) U (0, ∞)
X-Int: None
Y-Int: None
Rational Function
Asymptote: X = 0, Y = 0
End Behavior:
Top Curve:
As X--> ∞ , f(X)--> 0
As X--> 0 , f(X)--> ∞
Bottom Curve:
As X--> -∞, f(X)--> 0
As X--> 0, f(X)--> -∞
A Shift to the right 3
Equation: f(X) = 1/X-3
Domain: (-∞, 3) U (3, ∞)
Range: (-∞, 0) U (0, ∞)
X-Int: None
Y-Int: None
Asymptote: X = 3, Y = 0
End Behavior:
Top Curve:
As X--> ∞ , f(X)--> 0
As X--> 3 , f(X)--> ∞
Bottom Curve:
As X--> -∞, f(X)--> 0
As X--> 3, f(X)--> -∞
Translation Right
A shift down 5
Equation: f(X) = (1/X )- 5
Domain: (-∞, 0) U (0, ∞)
Range: (-∞, -5) U (-5, ∞)
X-Int: None
Y-Int: None
Asymptote: X = 0, Y = -5
End Behavior:
Top Curve:
As X--> ∞ , f(X)--> -5
As X--> 0 , f(X)--> ∞
Bottom Curve:
As X--> -∞, f(X)--> -5
As X--> 0, f(X)--> -∞
Translation Down
A vertical Flip - Reflection over the x-Axis
Equation: f(X) = -1/X
Domain: (-∞, 0) U (0, ∞)
Range: (-∞, 0) U (0, ∞)
X-Int: None
Y-Int: None
Asymptote: X = 0, Y = 0
End Behavior:
Top Curve:
As X--> -∞ , f(X)--> 0
As X--> 0 , f(X)--> ∞
Bottom Curve:
As X--> ∞, f(X)--> 0
As X--> 0, f(X)--> -∞
Reflection
A horizontal stretch by a factor of 2
Equation: f(X) = 2/X
Domain: (-∞, 0) U (0, ∞)
Range: (-∞, 0) U (0, ∞)
X-Int: None
Y-Int: None
End Behavior:
Top Curve:
As X--> ∞ , f(X)--> 0
As X--> 0 , f(X)--> ∞
Bottom Curve:
As X--> -∞, f(X)--> 0
As X--> 0, f(X)--> -∞
Horizontal
Stretch
Solving Rational Equations
Steps:
1. If necessary, factor out the denominators
2. Multiply fractions by LCD to get rid of them
3. Solve
4. Check solution to make sure it isn't extraneous (makes the denominator 0)
Equations & Steps
6/X+4 + 3X + 4/X^2-16 = 1/X -4
6/X+4 + 3x +4/(X-4)(X+4) = 1/X-4
LCD = (X-4)(X+4)
(X-4)( (X+4) (6/X+4) + (3X/(X+4)(X-4) = (1/X+4) )
6(X-4) + 3x + 4 = 1(X+4)
6X-24 + 3x +4 = X + 4
8X = 24
X = 3
CHECK WORK:
5/3X+3 + 8/9 = 28/36
5/3(X+1) + 8/9 = 28/36
LCD = 36(X+1)
(36(X+1) ( (5/3(X+1) + (8/9) = (28/36) )
5(12) + 8(4)(X+1) = 28(X+1)
60 + 32X + 32 = 28X + 28
4X = -64
X = -16
CHECK WORK:
Rational Function in the Real World
Real World Example
Roundabout
This is a roundabout. The curves, along the roundabout that are diagonal from each other, are a good example of a rational function. In the right corner there is an upwards curve and in the bottom left corner there is a downwards curve, which is similar to the shape of a rational function.