Questions and Answers
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 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)
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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)
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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)
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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)
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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)
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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)
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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)
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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?
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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)
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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?
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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)?
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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?
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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)
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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?
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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)
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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.
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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.