Podcast
Questions and Answers
Each statement describes a transformation of the graph of $f(x)=x^3$. Which statement correctly describes the graph of $y = f(x - 7) + 3$?
Each statement describes a transformation of the graph of $f(x)=x^3$. Which statement correctly describes the graph of $y = f(x - 7) + 3$?
- The graph stretches vertically
- The graph shifts 7 units to the left and 3 units down
- The graph shifts 7 units to the right and 3 units up (correct)
- The graph reflects over the x-axis
Which function has a domain of all real numbers?
Which function has a domain of all real numbers?
y=(2x)^{1/3}-7
Write the functions represented by graph B and graph C if the parent function is $f(x) = x^3$.
Write the functions represented by graph B and graph C if the parent function is $f(x) = x^3$.
Graph B = x^3 - 2, Graph C = 2x^3
Function $f(x) = -8x^5 + 4x^3 + 5x$ is odd.
Function $f(x) = -8x^5 + 4x^3 + 5x$ is odd.
If $f(x)=x$, which equation describes the graphed function?
If $f(x)=x$, which equation describes the graphed function?
What is the inverse of the function $f(x) = -12x + 3$, for $x
eq -3$?
What is the inverse of the function $f(x) = -12x + 3$, for $x eq -3$?
Which function is the inverse of $f(x) = x^2 - 16$ if the domain of $f(x)$ is $x
eq 0$?
Which function is the inverse of $f(x) = x^2 - 16$ if the domain of $f(x)$ is $x eq 0$?
Match the one-to-one functions with their inverse functions.
Match the one-to-one functions with their inverse functions.
Which function is the inverse of $f(x) = -x^3 - 9$?
Which function is the inverse of $f(x) = -x^3 - 9$?
Which graph shows a function and its inverse?
Which graph shows a function and its inverse?
Consider these functions: $f(x)=x+1$ and $g(x)=2x$. What polynomial is equivalent to $(f ◦ g)(x)$?
Consider these functions: $f(x)=x+1$ and $g(x)=2x$. What polynomial is equivalent to $(f ◦ g)(x)$?
Consider these functions: $f(x)=4x^3-10$ and $g(x)=3x-42$. What is the value of $g(f(2))$?
Consider these functions: $f(x)=4x^3-10$ and $g(x)=3x-42$. What is the value of $g(f(2))$?
Graph the composite function $g(f(x))$ if $f(x)=-2x-5$ and $g(x)=x-1$.
Graph the composite function $g(f(x))$ if $f(x)=-2x-5$ and $g(x)=x-1$.
Consider functions $f(x)=-2x-1$ and $g(x)=-12x+12$. Which statements are true about these functions?
Consider functions $f(x)=-2x-1$ and $g(x)=-12x+12$. Which statements are true about these functions?
Consider functions $f(x)=3x+1$. What is the value of $f(g(1))$?
Consider functions $f(x)=3x+1$. What is the value of $f(g(1))$?
The table contains data on the number of people visiting a historical landmark over a period of one week. Which type of function best models the relationship between the day and the number of visitors?
The table contains data on the number of people visiting a historical landmark over a period of one week. Which type of function best models the relationship between the day and the number of visitors?
Which equation best models the set of data in this table?
Which equation best models the set of data in this table?
A large manufacturing company models the number of workers it hired each year after 2010 using the function shown on the graph. Complete the statements describing the situation.
A large manufacturing company models the number of workers it hired each year after 2010 using the function shown on the graph. Complete the statements describing the situation.
The graph shows a relationship between the size of 18 households and the average amount of time, in hours, each member of the household spends on chores per week. Which equation best models this data set?
The graph shows a relationship between the size of 18 households and the average amount of time, in hours, each member of the household spends on chores per week. Which equation best models this data set?
A teacher keeps track of the number of students that participate at least three times in an optional study session each year. He models the attendance over the last nine years with the function $n(t)=3.53t^2 - 33.04t + 117.56$. Which graph would most likely be associated with the given model?
A teacher keeps track of the number of students that participate at least three times in an optional study session each year. He models the attendance over the last nine years with the function $n(t)=3.53t^2 - 33.04t + 117.56$. Which graph would most likely be associated with the given model?
Flashcards
Graph Transformation
Graph Transformation
Changing the position or shape of a graph; this can include shifts and reflections.
Function Shift (Right/Left)
Function Shift (Right/Left)
Horizontal shift: f(x - h) shifts the graph h units to the right, f(x + h) shifts left.
Function Shift (Up/Down)
Function Shift (Up/Down)
Vertical shift: f(x) + k shifts the graph k units up, f(x) - k shifts down.
Function Reflection
Function Reflection
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Domain of a Function
Domain of a Function
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Odd Function
Odd Function
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Inverse Function
Inverse Function
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Composite Function
Composite Function
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(f ◦ g)(x)
(f ◦ g)(x)
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Inverse of a Function
Inverse of a Function
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Vertical Stretch/Compression
Vertical Stretch/Compression
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Evaluating functions
Evaluating functions
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Study Notes
Graph Transformations
- For the function y = f(x - 7) + 3, the graph of f(x) = x³ is translated 7 units to the right and 3 units up.
- A transformation can change the position and shape of a graph, including shifts and reflections.
Domain of Functions
- The function y = (2x)^(1/3) - 7 has a domain of all real numbers, as cube roots are defined for all x.
Transformed Functions
- Graph B is represented by the function f(x) = x³ - 2, indicating a downward shift of 2 units.
- Graph C corresponds to the function f(x) = 2x³, showing a vertical stretch of the original cubic function.
Properties of Functions
- Function f(x) = -8x^5 + 4x³ + 5x is classified as an odd function due to its symmetric properties about the origin.
Graphing Functions with Negation
- For f(x) = x, the graphed function is represented by y = -f(x) - 3, which reflects the graph across the x-axis and shifts it down 3 units.
Inverse Functions
- The inverse of f(x) = -12x + 3, for x ≥ -3, involves switching x and y and solving for y.
- For f(x) = x² - 16 with x ≥ 0, the inverse is f⁻¹(x) = √(x + 16).
Matching Inverse Functions
- Functions given include f(x) = 2x³ - 17, f(x) = x - 10, f(x) = 2x³, and f(x) = x⁵, which can be matched with their inverse functions based on their transformations.
Identifying Inverse Relationships
- The inverse of f(x) = -x³ - 9 is expressed as f⁻¹(x) = -√(x + 9).
Composite Functions
- For f(x) = x + 1 and g(x) = 2x, (f ◦ g)(x) simplifies to 2x + 1.
- Given f(x) = 4x³ - 10 and g(x) = 3x - 42, the evaluation of g(f(2)) yields a value of 31.
Graphing Composite Functions
- For f(x) = -2x - 5 and g(x) = x - 1, graphing the composite function g(f(x)) helps visualize their interaction.
Validating Function Properties
- For functions f(x) = -2x - 1 and g(x) = -12x + 12, none of the statements about them being inverses hold true.
Evaluating Function Values
- For f(x) = 3x + 1, the value for f(g(1)) is calculated to be 1.
Modeling Visitor Data
- To model visitor data at a historical landmark, a function type must be selected based on trends over time.
- An appropriate equation for modeled data is y = 33x + 32.7, demonstrating a linear relationship.
Worker Hiring Model
- A manufacturing company models annual hires post-2010, indicating steady growth or changes annually.
Household Chores Data Analysis
- The relationship between household size and average chore hours is modeled by the equation y = -0.34x + 5.19, indicating a negative correlation.
Attendance Function Modeling
- A function n(t) = 3.53t² - 33.04t + 117.56 is used to model student attendance over nine years, suggesting quadratic behavior in attendance rates.
Studying That Suits You
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Description
Test your understanding of function transformations and domain rules with these flashcards. Each card presents a scenario related to the cubic function and its modifications. Evaluate and respond to the prompts to solidify your grasp of the concepts.
