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What are key features from graphs of quartic functions?
What are key features from graphs of quartic functions?
Lesson 15
Which statements are true about the function represented by the graph of f(x) with a zero at x = 3? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) with a zero at x = 3? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) with zeros at x = -3, x = 0, x = 1, and x = 2? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) with zeros at x = -3, x = 0, x = 1, and x = 2? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) with zeros at x = -2, x = 0.5, and x = 3? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) with zeros at x = -2, x = 0.5, and x = 3? (Select all that apply)
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What expression can represent a polynomial function given the x-intercepts are 0, -2, 1, and 7?
What expression can represent a polynomial function given the x-intercepts are 0, -2, 1, and 7?
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Which statements are true about the function represented by the graph of f(x) with a maximum of 0? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) with a maximum of 0? (Select all that apply)
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Which statements are true about the function represented by the graph of f(x) that has two relative minima and one relative maximum? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) that has two relative minima and one relative maximum? (Select all that apply)
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Which statements are true about the function represented by the graph of f(x) with one relative minimum and two relative maxima? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) with one relative minimum and two relative maxima? (Select all that apply)
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What is the zero of the function f(x) if it is x = -1?
What is the zero of the function f(x) if it is x = -1?
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Which statements are true about the function f(x) defined as f(x) = x(x - 2)(x + 6)(x + 1)? (Select all that apply)
Which statements are true about the function f(x) defined as f(x) = x(x - 2)(x + 6)(x + 1)? (Select all that apply)
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Which statements are true about the function f(x) defined as f(x) = -(x + 3)^4? (Select all that apply)
Which statements are true about the function f(x) defined as f(x) = -(x + 3)^4? (Select all that apply)
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Which statements are true about the function f(x) defined as f(x) = 5x^4 + 15x^3 - 20x^2 - 60x? (Select all that apply)
Which statements are true about the function f(x) defined as f(x) = 5x^4 + 15x^3 - 20x^2 - 60x? (Select all that apply)
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Which statements are true about the function f(x) defined as f(x) = x(x - 2)(x + 6)(x + 1)? (Select all that apply)
Which statements are true about the function f(x) defined as f(x) = x(x - 2)(x + 6)(x + 1)? (Select all that apply)
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If g(x) = f(x) - 1, then g(x) translates the function f(x) how many units?
If g(x) = f(x) - 1, then g(x) translates the function f(x) how many units?
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What happens to the graph when translated from f(x) = (x + 3)^4 to f'(x) = (x + 3)^4 + 11?
What happens to the graph when translated from f(x) = (x + 3)^4 to f'(x) = (x + 3)^4 + 11?
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Which graph best represents the translation of function f(x) = x^2 + 4x − 12 to function g(x) = x^2 + 4x − 4? (Select all that apply)
Which graph best represents the translation of function f(x) = x^2 + 4x − 12 to function g(x) = x^2 + 4x − 4? (Select all that apply)
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If g(x) = f(x - 1), then g(x) translates the function f(x) how many units?
If g(x) = f(x - 1), then g(x) translates the function f(x) how many units?
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What effect does the translation have on the function when changing from f(x) = x(x + 1)(x - 3) to g(x) = (x + 2)(x + 3)(x - 1) + 1?
What effect does the translation have on the function when changing from f(x) = x(x + 1)(x - 3) to g(x) = (x + 2)(x + 3)(x - 1) + 1?
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What happens to the graph when changing from f(x) = 2x^4 + x to g(x) = 2(x - 5)^4 + (x - 5)?
What happens to the graph when changing from f(x) = 2x^4 + x to g(x) = 2(x - 5)^4 + (x - 5)?
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Which function rule describes the transformation if g(x) = f(x) + k?
Which function rule describes the transformation if g(x) = f(x) + k?
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What is the value of k if g(x) = f(x) + k and the translation moves the graph down 11 units?
What is the value of k if g(x) = f(x) + k and the translation moves the graph down 11 units?
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Which function rule describes the transformation if the f(x) changes to g(x)? (where f(x) = -2x^4 - 2x^3 + 18x^2 + 18x)
Which function rule describes the transformation if the f(x) changes to g(x)? (where f(x) = -2x^4 - 2x^3 + 18x^2 + 18x)
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What happens to the graph of f(x) when it is transformed to g(x) = (-12x)^4 + 4 from f(x) = (6x)^4 + 4?
What happens to the graph of f(x) when it is transformed to g(x) = (-12x)^4 + 4 from f(x) = (6x)^4 + 4?
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Which graph best represents the dilation of function f(x) = x(x - 2)(x - 1) − 2 to function g(x) = 2x(x - 2)(x - 1) − 4? (Select all that apply)
Which graph best represents the dilation of function f(x) = x(x - 2)(x - 1) − 2 to function g(x) = 2x(x - 2)(x - 1) − 4? (Select all that apply)
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Study Notes
Key Features of Polynomial Functions
- Quartic functions have distinct characteristics on their graphs.
- Key graph features include x-intercepts, y-intercepts, domain, and range.
Graph Analysis and Characteristics
- The function f(x) demonstrates x-intercepts at specific points, which indicate the zeros of the function.
- The domain of polynomial functions is always (−∞, ∞).
- The range varies based on function behavior; for some functions, it may include maximum or minimum values.
- Analyzing the behavior of functions as they approach positive or negative infinity is crucial for understanding their overall shape.
Zeroes, Intercepts, and Roots
- Polynomial expressions can be expressed based on their x-intercepts.
- The expression can be factored to identify the zeros of the function, aiding in graph sketching.
Relative Minima and Maxima
- Functions may display relative maxima and minima indicating the highest or lowest points in specific intervals.
- These features help identify where the function increases or decreases.
Function Transformations
- Functions can be translated vertically or horizontally by adjusting their rules.
- Vertical translations are indicated by changes in the function's output, while horizontal shifts involve alterations in the input variable.
- Examples include translations of quadratic functions, where the vertex shifts in accordance with the changes.
Dilations of Functions
- Dilations can either stretch or compress the graph vertically or horizontally.
- Reflective properties occur when functions are multiplied by negative coefficients, changing their orientation across axes.
Real-World Applications of Polynomial Functions
- Understanding polynomial functions and their transformations is essential in various mathematical modeling scenarios.
- Features such as intercepts and extreme points serve not only in function analysis but also in practical applications in fields such as physics, engineering, and economics.
Summary of Specific Functions
- Certain functions demonstrate unique behaviors such as relative minima, maxima, and periodic peaks.
- Analyzing specific rules of functions, such as f(x) and g(x), provides insight into their transformations under dilation or translation.
Visual Representations
- Graphs play a vital role in visualizing function properties. Specific graphs are often referenced to illustrate transformations clearly, enhancing comprehension.
Importance of Analysis
- Distinguishing between different characteristics and behaviors of polynomial functions can greatly aid in calculus and higher-level mathematics.
- Familiarity with graph analysis, as well as the ability to predict changes due to transformations and dilations, is crucial for problem-solving in advanced mathematics.
Studying That Suits You
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Description
Test your understanding of key features of quartic functions and their graphs with this flashcard quiz. Explore important characteristics such as x-intercepts and y-intercepts through engaging questions and statements. Perfect for mastering polynomial function concepts in Algebra 2A.
