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Questions and Answers
What sequence of transformations correctly maps the graph of the function 𝑓 to the function 𝑔, where 𝑔(𝑥) = −𝑓(𝑥) + 5?
What sequence of transformations correctly maps the graph of the function 𝑓 to the function 𝑔, where 𝑔(𝑥) = −𝑓(𝑥) + 5?
- A vertical translation by 5 units, then a vertical reflection.
- A vertical reflection, followed by a vertical translation by 5 units. (correct)
- A vertical reflection, followed by a horizontal translation by 5 units.
- A horizontal translation by 5 units, then a vertical reflection.
After a vertical translation of the function 𝑓(𝑥) = −𝑥² + 3𝑥 + 2 by 4 units, which function represents the new graph?
After a vertical translation of the function 𝑓(𝑥) = −𝑥² + 3𝑥 + 2 by 4 units, which function represents the new graph?
- 𝑚(𝑥) = −(𝑥 + 4)² + 3(𝑥 + 4) + 2
- 𝑞(𝑥) = −𝑥² + 3𝑥 − 2
- 𝑝(𝑥) = −𝑥² + 3𝑥 + 6 (correct)
- 𝑛(𝑥) = −(𝑥 − 4)² + 3(𝑥 − 4) + 2
What effect does the operation of reflecting the function 𝑓 across the x-axis have on its graph?
What effect does the operation of reflecting the function 𝑓 across the x-axis have on its graph?
- It preserves the location of the vertex.
- It reverses the direction of the graph. (correct)
- It changes the concavity of the graph.
- It translates the graph horizontally.
Which of the following transformations is needed to convert 𝑓(𝑥) into a function that is vertically translated by 2 units?
Which of the following transformations is needed to convert 𝑓(𝑥) into a function that is vertically translated by 2 units?
What is the main difference between horizontal and vertical translations of a function?
What is the main difference between horizontal and vertical translations of a function?
What is the effect on the graph of the function when it is transformed to $g(x) = f(x) + 4$?
What is the effect on the graph of the function when it is transformed to $g(x) = f(x) + 4$?
Given the transformation $g(x) = f(x - 4)$, how does the graph of $g$ relate to the graph of $f$?
Given the transformation $g(x) = f(x - 4)$, how does the graph of $g$ relate to the graph of $f$?
What will be the output of $g(2)$ if $g(x) = f(x) + 2$ and $f(2) = 3$?
What will be the output of $g(2)$ if $g(x) = f(x) + 2$ and $f(2) = 3$?
If $f(x)$ has a domain of $[-4, 3]$, what will be the domain of $g(x) = -f(x + 5) + 2$?
If $f(x)$ has a domain of $[-4, 3]$, what will be the domain of $g(x) = -f(x + 5) + 2$?
How does the transformation $g(x) = -f(x)$ affect the original graph of $f$?
How does the transformation $g(x) = -f(x)$ affect the original graph of $f$?
If $g(x) = f(x - 2) + 3$, how is the range of $g$ compared to the range of $f$?
If $g(x) = f(x - 2) + 3$, how is the range of $g$ compared to the range of $f$?
Given $f(1) = -12$, what is $g(1)$ when $g(x) = f(x) - 4$?
Given $f(1) = -12$, what is $g(1)$ when $g(x) = f(x) - 4$?
What transformation is applied to $f(x)$ in the equation $g(x) = -f(x - 2) + 1$?
What transformation is applied to $f(x)$ in the equation $g(x) = -f(x - 2) + 1$?
What is the expression for 𝑔(𝑥) if 𝑓(𝑥) = 4𝑥 + 3?
What is the expression for 𝑔(𝑥) if 𝑓(𝑥) = 4𝑥 + 3?
If 𝑔(𝑥) is defined as 𝑔(𝑥) = 𝑓(𝑥 − 2) + 5, what does this transformation signify?
If 𝑔(𝑥) is defined as 𝑔(𝑥) = 𝑓(𝑥 − 2) + 5, what does this transformation signify?
For 𝑔(𝑥) = 𝑓(𝑥 + 3) + 4, what is the resulting domain if the original domain of 𝑓 is (0, 5)?
For 𝑔(𝑥) = 𝑓(𝑥 + 3) + 4, what is the resulting domain if the original domain of 𝑓 is (0, 5)?
What does the negative sign in 𝑔(𝑥) = −𝑓(𝑥) + 5 imply about the transformation of the graph of 𝑓?
What does the negative sign in 𝑔(𝑥) = −𝑓(𝑥) + 5 imply about the transformation of the graph of 𝑓?
What is the value of 𝑔(5) if 𝑔(𝑥) = 𝑓(𝑥) + 2 and 𝑓(5) = −2?
What is the value of 𝑔(5) if 𝑔(𝑥) = 𝑓(𝑥) + 2 and 𝑓(5) = −2?
If 𝑔(𝑥) = 𝑓(𝑥 − 2) + 4 and 𝑓(1) = 2, what is 𝑔(3)?
If 𝑔(𝑥) = 𝑓(𝑥 − 2) + 4 and 𝑓(1) = 2, what is 𝑔(3)?
If the range of the function 𝑓 is [−4, 8] and 𝑔(𝑥) = −𝑓(𝑥 + 3) + 2, what is the new range of 𝑔(𝑥)?
If the range of the function 𝑓 is [−4, 8] and 𝑔(𝑥) = −𝑓(𝑥 + 3) + 2, what is the new range of 𝑔(𝑥)?
Flashcards
Horizontal Shift
Horizontal Shift
A transformation of a function where the graph is shifted left or right.
Vertical Shift
Vertical Shift
A transformation of a function where the graph is shifted up or down.
Vertical Reflection
Vertical Reflection
A transformation of a function where the graph is flipped over the x-axis.
Combining Transformations
Combining Transformations
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Function notation (f(x + b))
Function notation (f(x + b))
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Function Notation(f(x) + k)
Function Notation(f(x) + k)
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Domain
Domain
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Range
Range
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Transforming graphs of functions
Transforming graphs of functions
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Vertical translation
Vertical translation
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Vertical reflection
Vertical reflection
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Horizontal translation
Horizontal translation
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Function transformation
Function transformation
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f(x) = -x^2 + 3x + 2 vertical translation by 4
f(x) = -x^2 + 3x + 2 vertical translation by 4
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g(x) = -f(x) + 5
g(x) = -f(x) + 5
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g(x) = f(x - 5) - 3
g(x) = f(x - 5) - 3
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g(x) = f(x) - 4
g(x) = f(x) - 4
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g(x) = -f(x - 3) + 1
g(x) = -f(x - 3) + 1
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f(x) = 4x + 3
f(x) = 4x + 3
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f(x) = 2x - 5
f(x) = 2x - 5
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g(x) = f(x) + 5
g(x) = f(x) + 5
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g(x) = f(x + 3) + 4
g(x) = f(x + 3) + 4
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f(x) = x³ + 2x²
f(x) = x³ + 2x²
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f(x) = 2x² - 3x + 1
f(x) = 2x² - 3x + 1
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g(x) = -f(x) + 5
g(x) = -f(x) + 5
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g(x) = f(x - 2) + 5
g(x) = f(x - 2) + 5
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Study Notes
Translations of Functions
- Graphic Transformations: Given a graph of a function (f), translations can be applied graphically to create a new graph (g).
- Horizontal Shifts: Shifting the graph of f(x) horizontally by 'a' units results in g(x) = f(x - a) (left 'a' units) or g(x) = f(x + a) (right 'a' units).
- Vertical Shifts: Shifting the graph of f(x) vertically by 'a' units results in g(x) = f(x) + a (up 'a' units) or g(x) = f(x) - a (down 'a' units).
- Vertical Reflection: Reflecting the graph of f(x) across the x-axis creates g(x) = -f(x).
Additive Transformations (Algebraically)
- Example 4: Given f(x) = x² - 3x + 2, find g(x) if g(x) = f(x) + 4. (This involves a vertical translation of 4 units up)
- Example 4 Alternative (using a table): Given a table of values for f(x), find g(2) if g(x) = f(x) + 2.
- Example 4 Alternative (translation): Given a table of values for f(x), find g(y) if g(x) = f(x – 2) + 1.
Numerical Transformations
- Example 5: Using a table of values for f(x), find values for g(x) if g(x) = f(x) + 2 or if g(x) = f(x – 2) + 1.
Domain and Range
- Example 6: Given the graph of f with a domain of [-4, 3] and a range of (3, 9), find the domain and range of g(x) = -f(x + 5) + 2.
- Determining domain and range changes based on how the graph is being transformed.
Practice Problems
- Graphics Transformations: Use the graph of f(x) to graph g(x) for various equations involving translations like: g(x) = f(x - 2) + 4, g(x) = f(x + 3), etc.
- Algebraic Transformations: Rewrite g(x) in terms of x. Examples: Given f(x) = 4x + 3, find g(x) = f(x) + 5.
Additional Practice (Numeric Transf)
- Given a table with x and f(x) values, calculate g(x) for g(x) = f(x) + a, g(x) = f(x + a), other relevant transformations.
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