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SECTION 2-6Transformations of Functions
ESSENTIAL QUESTIONS• How do you identify the effects on the graphs of
functions by replacing f(x) with f(x) + k and f(x - h) for positive and negative values?
• How do you identify the effect on the graphs of functions by replacing f(x) with af(x), f(ax), -af(x), and f(-ax)?
VOCABULARY1. Parent Graph:
2. Parent Function:
3. Transformation:
4. Translation:
5. Reflection:
VOCABULARY1. Parent Graph: The simplest version of a graph of a
function2. Parent Function:
3. Transformation:
4. Translation:
5. Reflection:
VOCABULARY1. Parent Graph: The simplest version of a graph of a
function2. Parent Function: The function that generates the parent
graph
3. Transformation:
4. Translation:
5. Reflection:
VOCABULARY1. Parent Graph: The simplest version of a graph of a
function2. Parent Function: The function that generates the parent
graph
3. Transformation: When a graph is slid/shifted, reflected, stretched, or compressed
4. Translation:
5. Reflection:
VOCABULARY1. Parent Graph: The simplest version of a graph of a
function2. Parent Function: The function that generates the parent
graph
3. Transformation: When a graph is slid/shifted, reflected, stretched, or compressed
4. Translation: When a graph is moved by a horizontal and/or vertical shift
5. Reflection:
VOCABULARY1. Parent Graph: The simplest version of a graph of a
function2. Parent Function: The function that generates the parent
graph
3. Transformation: When a graph is slid/shifted, reflected, stretched, or compressed
4. Translation: When a graph is moved by a horizontal and/or vertical shift
5. Reflection: When a graph flips across a line
VOCABULARY6. Line of Reflection:
7. Dilation:
VOCABULARY6. Line of Reflection: The line that a graph reflects across
7. Dilation:
VOCABULARY6. Line of Reflection: The line that a graph reflects across
7. Dilation: When a graph changes is size but not in shape
PARENT GRAPHS
x
y
y = x
PARENT GRAPHS
x
y
y = x
PARENT GRAPHS
x
y
y = x
PARENT GRAPHS
x
y
y = x
PARENT GRAPHS
x
y
y = x 2
PARENT GRAPHS
x
y
y = x 2
Transformation
Transformation Change to Parent Graph f(x)
Transformation Change to Parent Graph f(x)Horizontal
Translation h
Transformation Change to Parent Graph f(x)Horizontal
Translation hf(x + h) shifts h units leftf(x - h) shifts h units right
Transformation Change to Parent Graph f(x)Horizontal
Translation hf(x + h) shifts h units leftf(x - h) shifts h units right
Vertical Translation k
Transformation Change to Parent Graph f(x)Horizontal
Translation hf(x + h) shifts h units leftf(x - h) shifts h units right
Vertical Translation k
f(x) + k shifts k units upf(x) - k shifts k units down
Transformation Change to Parent Graph f(x)Horizontal
Translation hf(x + h) shifts h units leftf(x - h) shifts h units right
Vertical Translation k
f(x) + k shifts k units upf(x) - k shifts k units down
Reflection
Transformation Change to Parent Graph f(x)Horizontal
Translation hf(x + h) shifts h units leftf(x - h) shifts h units right
Vertical Translation k
f(x) + k shifts k units upf(x) - k shifts k units down
Reflection -f(x) reflects over the x-axisf(-x) reflects over the y-axis
Transformation Change to Parent Graph f(x)Horizontal
Translation hf(x + h) shifts h units leftf(x - h) shifts h units right
Vertical Translation k
f(x) + k shifts k units upf(x) - k shifts k units down
Reflection -f(x) reflects over the x-axisf(-x) reflects over the y-axis
Dilation a
Transformation Change to Parent Graph f(x)Horizontal
Translation hf(x + h) shifts h units leftf(x - h) shifts h units right
Vertical Translation k
f(x) + k shifts k units upf(x) - k shifts k units down
Reflection -f(x) reflects over the x-axisf(-x) reflects over the y-axis
Dilation a
f(ax) compresses horizontally when |a| > 0
f(ax) stretches horizontally when 0 < |a| < 1
a･f(x) stretches vertically when |a| > 1
a･f(x) compresses vertically when 0 < |a| < 1
EXAMPLE 1Describe the translation as it relates to its parent
graph. Then graph the function.
y = (x +1)2
x
y
EXAMPLE 1Describe the translation as it relates to its parent
graph. Then graph the function.
y = (x +1)2
Shift one unit left from the original parent function
y = x2x
y
EXAMPLE 1Describe the translation as it relates to its parent
graph. Then graph the function.
y = (x +1)2
Shift one unit left from the original parent function
y = x2x
y
EXAMPLE 2Describe the reflection as it relates to its parent
graph. Then graph the function.
y = − x
x
y
EXAMPLE 2Describe the reflection as it relates to its parent
graph. Then graph the function.
y = − xReflect the parent function
over the x-axisy = x
x
y
EXAMPLE 2Describe the reflection as it relates to its parent
graph. Then graph the function.
y = − xReflect the parent function
over the x-axisy = x
x
y
EXAMPLE 3Describe the dilation as it relates to its parent graph.
Then graph the function.
y = 12x
x
y
EXAMPLE 3Describe the dilation as it relates to its parent graph.
Then graph the function.
y = 12x
Compress the parent function vertically by half
y = x
x
y
EXAMPLE 3Describe the dilation as it relates to its parent graph.
Then graph the function.
y = 12x
Compress the parent function vertically by half
y = x
x
y
EXAMPLE 4The graph of f(x) is shown on the grid. If the graph of f(x) is translated 4 units to the left, 1 unit up, and reflected over the x-axis to create the graph of
g(x), write a function that best represents g(x) in terms of f(x).
x
y
EXAMPLE 4The graph of f(x) is shown on the grid. If the graph of f(x) is translated 4 units to the left, 1 unit up, and reflected over the x-axis to create the graph of
g(x), write a function that best represents g(x) in terms of f(x).
x
y
Original:
EXAMPLE 4The graph of f(x) is shown on the grid. If the graph of f(x) is translated 4 units to the left, 1 unit up, and reflected over the x-axis to create the graph of
g(x), write a function that best represents g(x) in terms of f(x).
x
y
Original: g(x) = -f(x - 3) + 8
EXAMPLE 4The graph of f(x) is shown on the grid. If the graph of f(x) is translated 4 units to the left, 1 unit up, and reflected over the x-axis to create the graph of
g(x), write a function that best represents g(x) in terms of f(x).
x
y
Original: g(x) = -f(x - 3) + 8 g(x) = x − 3 + 8
EXAMPLE 4The graph of f(x) is shown on the grid. If the graph of f(x) is translated 4 units to the left, 1 unit up, and reflected over the x-axis to create the graph of
g(x), write a function that best represents g(x) in terms of f(x).
x
y
g(x) = f(x + 1) - 9 Original: g(x) = -f(x - 3) + 8 g(x) = x − 3 + 8
EXAMPLE 4The graph of f(x) is shown on the grid. If the graph of f(x) is translated 4 units to the left, 1 unit up, and reflected over the x-axis to create the graph of
g(x), write a function that best represents g(x) in terms of f(x).
x
y
g(x) = f(x + 1) - 9 g(x) = x +1− 9
Original: g(x) = -f(x - 3) + 8 g(x) = x − 3 + 8
