GPS Algebra
Function Transformations
ABSOLUTE VALUE FUNCTION TRANSFORMATIONS
The following standard form is used for Absolute Value functions: f(x) = a|x – h| + k
a – leading coefficient; affects whether the function opens up or down; also affects the graph’s steepness
(h, k) – vertex of the graph; determines up/down and left/right translations
Addition inside absolute value (h)  translate left
Subtraction inside absolute value (h)  translate right
Addition outside absolute value (k)  translate up
Subtraction outside absolute value (k)  translate down
Steps:
1)
2)
3)
4)
5)
Examine the function and write all transformations
Check sign of leading coefficient (a) – positive opens up, negative opens down
Use (h, k) to find and graph the vertex (reminder: use the opposite sign for h)
Use the number value of a to graph the sides of the graph (similar to linear slope)
Use the graph to determine domain and range
Ex. 1)
f(x) = 2| x |
Ex. 2)
f(x) = – ½ | x |
Domain: __________
Domain: __________
Range:
Range:
Ex. 3)
__________
f(x) = –3| x + 2 | – 1
Ex. 4)
__________
f(x) = ¼ | x – 1 | + 3
Domain: __________
Domain: __________
Range:
Range:
__________
__________
GPS Algebra
Function Transformations
QUADRATIC FUNCTION TRANSFORMATIONS
The following standard form is used for Quadratic functions: f(x) = a(x – h)2 + k
a – leading coefficient; affects whether the function opens up or down; also affects the graph’s steepness
(h, k) – vertex of the graph; determines up/down and left/right translations
Addition inside binomial (h)  translate left
Subtraction inside binomial (h)  translate right
Addition outside binomial (k)  translate up
Subtraction outside binomial (k)  translate down
Steps:
1)
2)
3)
4)
5)
Examine the function and write all transformations
Check sign of leading coefficient (a) – positive opens up, negative opens down
Use (h, k) to find and graph the vertex (reminder: use the opposite sign for h)
Use the number value of a to graph the curved sides of the parabola
Use the graph to determine domain and range
Ex. 1)
f(x) = 2(x)2
Ex. 2)
f(x) = – ½ (x)2
Domain: __________
Domain: __________
Range:
Range:
Ex. 3)
__________
f(x) = –3(x + 2)2 – 1
Ex. 4)
__________
f(x) = ¼ (x – 1)2 + 3
Domain: __________
Domain: __________
Range:
Range:
__________
__________
GPS Algebra
Function Transformations
CUBIC FUNCTION TRANSFORMATIONS
The following standard form is used for Cubic functions: f(x) = a(x – h)3 + k
a – leading coefficient; affects whether the function opens up or down; also affects the graph’s steepness
(h, k) – vertex of the graph; determines up/down and left/right translations
Addition inside binomial (h)  translate left
Subtraction inside binomial (h)  translate right
Addition outside binomial (k)  translate up
Subtraction outside binomial (k)  translate down
Steps:
1)
2)
3)
4)
5)
Examine the function and write all transformations
Check sign of leading coefficient (a) – positive opens up, negative opens down
Use (h, k) to find and graph the vertex (reminder: use the opposite sign for h)
Use the number value of a to graph the curved sides of the curve
Use the graph to determine domain and range
Ex. 1)
f(x) = 2(x)3
Ex. 2)
f(x) = – ½ (x)3
Domain: __________
Domain: __________
Range:
Range:
Ex. 3)
__________
f(x) = –3(x + 2)3 – 1
Ex. 4)
__________
f(x) = ¼ (x – 1)3 + 3
Domain: __________
Domain: __________
Range:
Range:
__________
__________
GPS Algebra
Function Transformations
SQUARE ROOT FUNCTION TRANSFORMATIONS
The following standard form is used for Square Root functions: f(x) = a√(𝒙 – 𝒉) + k
a – leading coefficient; affects whether the function opens up and over or down and over; also affects the
graph’s steepness
(h, k) – vertex of the graph; determines up/down and left/right translations
Addition inside radical (h)  translate left
Subtraction inside radical (h)  translate right
Addition outside radical (k)  translate up
Subtraction outside radical (k)  translate down
Steps:
Ex. 1)
1) Examine the function and write all transformations
2) Check sign of leading coefficient (a) – positive opens up and over, negative opens down
and over
3) Use (h, k) to find and graph the endpoint (reminder: use the opposite sign for h)
4) Use the number value of a to graph the curved sides of the parabola
5) Use the graph to determine domain and range
f(x) = 2√𝑥
Ex. 2)
f(x) = – ½ √𝑥
Domain: __________
Domain: __________
Range:
Range:
Ex. 3)
__________
f(x) = –3√𝑥 + 2 – 1
Ex. 4)
__________
f(x) = ¼ √𝑥 – 1 + 3
Domain: __________
Domain: __________
Range:
Range:
__________
__________
GPS Algebra
Function Transformations
RATIONAL FUNCTION TRANSFORMATIONS
𝒂
The following standard form is used for Rational functions: f(x) = 𝒙−𝒉 + 𝒌
a – coefficient in numerator; affects whether the function is reflected down; also affects the graph’s steepness
(h, k) – center of the graph; determines up/down and left/right translations
Addition in denominator (h)  translate left
Subtraction in denominator (h)  translate right
Addition in denominator (k)  translate up
Subtraction in denominator (k)  translate down
Steps:
Ex. 1)
1) Examine the function and write all transformations
2) Check sign of leading coefficient (a) – positive keeps parent function shape and negative
is reflected down
3) Use (h, k) to find and locate the center (reminder: use the opposite sign for h)
4) Extend up/down and left/right from the center for asymptotes
5) Locate anchor points and sketch graph
6) Use the graph to determine domain and range
1
f(x) = 𝑥−2
Ex. 2)
1
f(x) = 𝑥 − 2
Domain: __________
Domain: __________
Range:
Range:
Ex. 3)
__________
1
f(x) = – 𝑥+2
Ex. 4)
__________
1
f(x) = 𝑥−1 + 2
Domain: __________
Domain: __________
Range:
Range:
__________
__________