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Questions and Answers
What is the result of applying a vertical dilation of 4 and a vertical translation by 6 units to the graph of the function 𝑓?
What is the result of applying a vertical dilation of 4 and a vertical translation by 6 units to the graph of the function 𝑓?
- The y-values are multiplied by 4 and the x-values are unaffected.
- The graph is reflected across the x-axis and translated down.
- The y-values are only increased by 6.
- The y-values are multiplied by 4 and then increased by 6. (correct)
Given the function 𝑔 is defined as 𝑔(𝑥) = 3𝑓(2𝑥) + 𝑏, what effect does the '2' have on the function 𝑓?
Given the function 𝑔 is defined as 𝑔(𝑥) = 3𝑓(2𝑥) + 𝑏, what effect does the '2' have on the function 𝑓?
- It compresses the graph horizontally by a factor of 2. (correct)
- It stretches the graph horizontally by a factor of 2.
- It flips the graph over the y-axis.
- It translates the graph upward by 2 units.
What does finding the zeroes of the function 𝑔 indicate?
What does finding the zeroes of the function 𝑔 indicate?
- The maximum points of the function 𝑔.
- The x-values where the graph touches the y-axis.
- The x-values where 𝑔(𝑥) = 0. (correct)
- The y-values where the graph crosses the x-axis.
If the domain of a function is restricted, which of the following implications might it have?
If the domain of a function is restricted, which of the following implications might it have?
What represents the y-intercept of the function 𝑔 when evaluated at 𝑔(0)?
What represents the y-intercept of the function 𝑔 when evaluated at 𝑔(0)?
What effect does the function transformation $g(x) = 2f(x)$ have on the graph of $f(x)$?
What effect does the function transformation $g(x) = 2f(x)$ have on the graph of $f(x)$?
If the function $g(x) = f(2x)$ is applied, what transformation is performed on the graph of $f(x)$?
If the function $g(x) = f(2x)$ is applied, what transformation is performed on the graph of $f(x)$?
What is the domain of the transformed function $g(x) = 3f(x) + 2$ if the domain of $f(x)$ is $[2, 5]$?
What is the domain of the transformed function $g(x) = 3f(x) + 2$ if the domain of $f(x)$ is $[2, 5]$?
For the function $g(x) = f(x) - 3$, how does the range of $g(x)$ compare to the range of $f(x)$ if $f(x)$ has a range of $[4, 9]$?
For the function $g(x) = f(x) - 3$, how does the range of $g(x)$ compare to the range of $f(x)$ if $f(x)$ has a range of $[4, 9]$?
If you have a function $f(x)$ with a zero at $x = 1$, what can be said about the function $g(x) = f(3x - 2)$?
If you have a function $f(x)$ with a zero at $x = 1$, what can be said about the function $g(x) = f(3x - 2)$?
Given the function $f(x)$ has intercepts at $(0, 20)$ and $(2, 0)$, what can be inferred about the function $g(x) = 5f(x)$?
Given the function $f(x)$ has intercepts at $(0, 20)$ and $(2, 0)$, what can be inferred about the function $g(x) = 5f(x)$?
If the function $g(x) = f(2x + 1) - 4$, what is the new position of the horizontal shift?
If the function $g(x) = f(2x + 1) - 4$, what is the new position of the horizontal shift?
What transformation is represented by the function $g(x) = f(x - 3) + 5$?
What transformation is represented by the function $g(x) = f(x - 3) + 5$?
What is the effect of the transformation $g(x) = 3f(x) + 5$ on the graph of $f(x)$?
What is the effect of the transformation $g(x) = 3f(x) + 5$ on the graph of $f(x)$?
If $g(x) = f(3x) - 1$, what type of transformation does this represent for the function $f(x)$?
If $g(x) = f(3x) - 1$, what type of transformation does this represent for the function $f(x)$?
If $g(x) = f(x) + 4$, what is the domain of $g(x)$ if the domain of $f(x)$ is $ ext{[a, b]}$?
If $g(x) = f(x) + 4$, what is the domain of $g(x)$ if the domain of $f(x)$ is $ ext{[a, b]}$?
Which transformation does $g(x) = f(x - 2) + 3$ represent?
Which transformation does $g(x) = f(x - 2) + 3$ represent?
For which of these functions does the transformation create a vertical reflection?
For which of these functions does the transformation create a vertical reflection?
If $f(x)$ has a zero at $x = 3$, what can be said about $g(x) = f(2x - 6)$?
If $f(x)$ has a zero at $x = 3$, what can be said about $g(x) = f(2x - 6)$?
What is the range of the function $g(x) = 2f(x) + 1$ if the original range of $f(x)$ is $[c, d]$?
What is the range of the function $g(x) = 2f(x) + 1$ if the original range of $f(x)$ is $[c, d]$?
If the domain of $f(x)$ is $[0, 5]$, what is the domain of $g(x) = f(x + 1)$?
If the domain of $f(x)$ is $[0, 5]$, what is the domain of $g(x) = f(x + 1)$?
Flashcards
Transforming functions
Transforming functions
Applying changes (like stretching, shrinking, shifts) to a function's graph.
Horizontal Dilation
Horizontal Dilation
Stretching or compressing a graph horizontally, by multiplying x values by a constant.
Vertical Dilation
Vertical Dilation
Stretching or compressing a graph vertically, by multiplying y values by a constant.
Vertical Translation
Vertical Translation
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Function Composition
Function Composition
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𝑔 𝑥 = 3𝑓 2𝑥 + 𝑏
𝑔 𝑥 = 3𝑓 2𝑥 + 𝑏
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Finding g(4)
Finding g(4)
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Zeroes of a Function
Zeroes of a Function
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y-intercept
y-intercept
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Dilating a function
Dilating a function
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Vertical Dilation
Vertical Dilation
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Horizontal Dilation
Horizontal Dilation
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𝑓(3𝑥) transformation
𝑓(3𝑥) transformation
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Dilations of functions
Dilations of functions
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𝑓(𝑥) + c
𝑓(𝑥) + c
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𝑓(x+c)
𝑓(x+c)
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Multiplicative transformations
Multiplicative transformations
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2𝑓(𝑥)
2𝑓(𝑥)
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g(x) = 2f(x)
g(x) = 2f(x)
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g(x) = f(2x)
g(x) = f(2x)
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Domain of a function
Domain of a function
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Range of a function
Range of a function
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g(x) = 3f(x) + 2
g(x) = 3f(x) + 2
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Domain
Domain
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Horizontal transformation
Horizontal transformation
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Vertical transformation
Vertical transformation
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Range
Range
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𝑓(2𝑥)
𝑓(2𝑥)
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Study Notes
Dilations of Functions
- Dilations are multiplicative transformations.
- Vertical Dilations:
- Given a graph f(x), if g(x) = af(x), the graph of g(x) is a vertical stretch or compression of f(x) by a factor of |a|. If a > 1, it's a vertical stretch. If 0 < a < 1, it's a vertical compression. If a < 0, there's a vertical stretch or compression, and a reflection over the x-axis.
- Horizontal Dilations:
- If g(x) = f(bx), the graph of g(x) is a horizontal stretch or compression of f(x) by a factor of 1/|b|. If b > 1, it's a horizontal compression. If 0 < b < 1, it's a horizontal stretch. If b < 0, there's a horizontal compression or stretch, and a reflection over the y-axis.
- Combined Transformations: Transformations can be combined.
- Order matters: Horizontal transformations are applied before vertical transformations in the function.
- Example: g(x) = a f(b(x – c)) + d where "a" affects the vertical scale, "b" affects the horizontal scale, "c" shifts the graph horizontally, and "d" shifts the graph vertically.
Algebraic Transformations
- Given a function, find another function based on transformations of the original.
- Example: if f(x) = x² – 3x + 2, and g(x) = 2f(x) + 4, what is g(x)?
- The transformation of the function is expressed in terms of x.
Numerical Transformations
- Transformations of functions based on data represented in a table.
- Example: If a table of values for f(x) is given, and g(x) = af(x) +d, evaluate g(x) at a specific point.
Domain and Range Transformations
- Given the domain and range of a function, determine the domain and range of a transformation.
- Examples: How do horizontal and vertical dilations affect the domain and range?
- If f(x) has domain [a, b] and a range [c, d], determine the domain and range of g(x) = af(bx + c) + d.
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