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Questions and Answers
Given $f(x)$ and $g(x) = f(x - 4) + 3$, what transformations have been applied to $f(x)$ to obtain $g(x)$?
Given $f(x)$ and $g(x) = f(x - 4) + 3$, what transformations have been applied to $f(x)$ to obtain $g(x)$?
- Shifted 3 units to the left and 4 units down.
- Shifted 4 units to the right and 3 units up. (correct)
- Shifted 4 units to the left and 3 units down.
- Shifted 3 units to the right and 4 units up.
Given the function $f(x)$, which transformation results in a horizontal stretch by a factor of 2 and a reflection across the y-axis?
Given the function $f(x)$, which transformation results in a horizontal stretch by a factor of 2 and a reflection across the y-axis?
- $-f(\frac{1}{2}x)$
- $f(-2x)$
- $\frac{1}{2}f(-x)$
- $f(-\frac{1}{2}x)$ (correct)
A function $p(x)$ has a known value at $x = a$. If $h(x) = 3p(2x) - 1$, at what value must we know $p(x)$ to determine $h(a)$?
A function $p(x)$ has a known value at $x = a$. If $h(x) = 3p(2x) - 1$, at what value must we know $p(x)$ to determine $h(a)$?
- We must know $p(a/2)$.
- We must know $p(a)$.
- We must know $p(2a)$. (correct)
- We must know $p(3a-1)$.
A function $f(x)$ is transformed to $g(x) = af(x + h) + k$. If $|a| > 1$ and $h > 0$, describe the transformations applied to $f(x)$.
A function $f(x)$ is transformed to $g(x) = af(x + h) + k$. If $|a| > 1$ and $h > 0$, describe the transformations applied to $f(x)$.
If $h(x) = ap(bx) + c$, what is the effect of changing the value of 'b' on the graph of $p(x)$?
If $h(x) = ap(bx) + c$, what is the effect of changing the value of 'b' on the graph of $p(x)$?
The graph of $f(x) = x^3$ is transformed to $g(x) = (x - 2)^3 + 1$. Describe the transformations.
The graph of $f(x) = x^3$ is transformed to $g(x) = (x - 2)^3 + 1$. Describe the transformations.
A model predicts a distance of 68 meters, but the actual distance measured is 70 meters. What is the residual, and what does it indicate?
A model predicts a distance of 68 meters, but the actual distance measured is 70 meters. What is the residual, and what does it indicate?
Given the function $d(T) = 2T + 10$ that models the distance based on temperature $T$, which statement best describes the model's limitation?
Given the function $d(T) = 2T + 10$ that models the distance based on temperature $T$, which statement best describes the model's limitation?
Consider a function $f(x)$. Which transformation affects the domain of $f(x)$ but not necessarily the range?
Consider a function $f(x)$. Which transformation affects the domain of $f(x)$ but not necessarily the range?
The number of bacteria $N(T)$ at temperature $T$ is modeled by $N(T) = 30T^2 - 30T + 120$, where $-2 < T < 14$. Why is it inappropriate to use this model to predict the number of bacteria at $T = -16^\circ C$?
The number of bacteria $N(T)$ at temperature $T$ is modeled by $N(T) = 30T^2 - 30T + 120$, where $-2 < T < 14$. Why is it inappropriate to use this model to predict the number of bacteria at $T = -16^\circ C$?
Given $f(x) = |x|$, what is the equation of the transformed function after a horizontal compression by a factor of 3 and a reflection across the y-axis?
Given $f(x) = |x|$, what is the equation of the transformed function after a horizontal compression by a factor of 3 and a reflection across the y-axis?
Where does the parabola $f(x) = x^2$ have its minimum value?
Where does the parabola $f(x) = x^2$ have its minimum value?
A bacterial population is modeled by the function $P(t)$, where $t$ is measured in hours. If the growth rate is slowed down by a factor of 2, and the initial population is tripled, which transformation represents the new population model?
A bacterial population is modeled by the function $P(t)$, where $t$ is measured in hours. If the growth rate is slowed down by a factor of 2, and the initial population is tripled, which transformation represents the new population model?
Suppose a regression equation models the relationship between temperature ($x$) and ice cream sales ($y$). A transformation to the temperature data is applied such that $x$ is replaced with $x + 5$. How does this transformation affect the interpretation of the model?
Suppose a regression equation models the relationship between temperature ($x$) and ice cream sales ($y$). A transformation to the temperature data is applied such that $x$ is replaced with $x + 5$. How does this transformation affect the interpretation of the model?
Given $g(x) = (x - 4)^2 + 3$, what is the range of the function?
Given $g(x) = (x - 4)^2 + 3$, what is the range of the function?
Which of the following transformations to $f(x)$ will change both its x and y intercepts?
Which of the following transformations to $f(x)$ will change both its x and y intercepts?
Flashcards
Vertical Translation
Vertical Translation
Shifts the graph up or down by 'k' units.
Horizontal Translation
Horizontal Translation
Shifts the graph left (h > 0) or right (h < 0).
Vertical Dilation
Vertical Dilation
Stretches/compresses vertically by a factor of |a|. Reflects over x-axis if a < 0.
Horizontal Dilation
Horizontal Dilation
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f(x - 3)
f(x - 3)
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g(x) = -f(x)
g(x) = -f(x)
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If f(x) = x^2, vertical dilation by 3 & horizontal translation 2 left
If f(x) = x^2, vertical dilation by 3 & horizontal translation 2 left
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the sequence of transformations from f(x) to h(x) = 2f(3x) - 1
the sequence of transformations from f(x) to h(x) = 2f(3x) - 1
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Horizontal Stretch/Compression
Horizontal Stretch/Compression
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Vertical Stretch/Compression
Vertical Stretch/Compression
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Vertical Shift
Vertical Shift
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Horizontal Shift
Horizontal Shift
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Domain
Domain
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Range
Range
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Residual
Residual
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Extrapolation
Extrapolation
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Study Notes
Transformations of Functions
- Vertical Translation: g(x) = f(x) + k, shifts the graph up or down by k units.
- Horizontal Translation: g(x) = f(x + h), shifts the graph left if h > 0 and right if h < 0.
- Vertical Dilation: g(x) = af(x), stretches or compresses vertically by a factor of a, reflecting over the x-axis if a < 0.
- Horizontal Dilation: g(x) = f(bx), stretches or compresses horizontally by a factor of 1/b, reflecting over the y-axis if b < 0.
Function Model Selection and Assumption Articulation
- Includes selecting appropriate models (linear, quadratic, etc.) for real-world scenarios.
- Includes interpreting regression equations, correlation coefficients (r), and residuals.
Function Model Construction and Application
- Construct models using transformations.
- Apply models to solve contextual problems like temperature vs. fish distance and bacterial growth.
Examples of Parent Functions and Their Transformations
- Constant, Identity, Quadratic, Cubic, Rational, Square Root, Cube Root, and Absolute Value functions.
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Description
Learn about vertical/horizontal translations and dilations of functions. Construct models and solve contextual problems using these transformations. Includes interpreting regression equations, correlation coefficients (r), and residuals.
