Questions and Answers
Which statements are true about the function represented by the graph of f(x)? (Select all that apply)
Which statements are true about the function represented by the graph of f(x) from question 1? (Select all that apply)
Which statements are true about the function f(x) = (x + 4)^2(x − 2)^2? (Select all that apply)
Which statements are true about the function f(x) = (x + 4)^2(x - 2)^2 from question 3? (Select all that apply)
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Which graph represents the function that has the rule f(x) = (x + 4)^2(x − 2)^2?
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What describes how the graph of f(x) changed to g(x) = x^4 + 2x^3 + 4?
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Which graph best represents the translation of function f(x) = x^3 + 5x^2 to function g(x) = x^3 + 5x^2 - 4?
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What describes how the graph of f(x) changed to g(x) = (x + 3)^2 + 8?
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If the rule of the function for f(x) is f(x) = x^2 − 4, what is the rule of the function for g(x)?
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What describes how the graph of f(x) changed to g(x) = 2(x − 3)^2 + 4?
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What describes how the graph of f(x) changed to g(x) = 3x^4 + 1?
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What describes how the graph changed from f(x) = -2(x - 1)^4 + 7 to g(x) = (x - 1)^4 - 72?
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Which graph best represents the translation of function f(x) = x(x − 2)^2 − 2 to function g(x) = 3x(x − 2)^2 − 6?
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Which graph best represents the translation of function f(x) = (−2x)^3 + 5 to function g(x) = (2x)^3 + 5?
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What describes how the graph of f(x) changed to g(x) = (9x)^3 − 8?
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What describes how the graph of f(x) changed to g(x) = (12x)^4 + 4?
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Which graph best represents the translation of function f(x) = −x^2(x − 1)(x + 2) + 5 to function g(x) = (2x)^2(2x − 1)(2x + 2) − 5?
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If the rule of the function is f(x) = (x − 2)^3 + 3, what is the rule of the function for g(x)?
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Study Notes
Function Properties and Graphs
- Zeros of a function indicate values where the function intersects the x-axis. For a given function, the zeros are x = -3 and x = 1.5.
- X-intercepts and the y-intercept define points where the function crosses the axes: x-intercepts are (-3, 0), (1.5, 0); y-intercept is (0, 20.3).
- The domain of a function typically includes all real numbers (−∞, ∞), while the range indicates the valid output values, here [0, ∞).
Relative Extrema and Behavior
- A function can have multiple relative minima and maxima; in this case, it has two minima at 0 and one maximum at 25.6.
- Function behavior can be characterized by how it approaches infinity: as x approaches both negative and positive infinity, f(x) approaches infinity.
- Increasing and decreasing intervals help describe function behavior: it increases on (-3, −0.7) and (1.5, ∞) and decreases on (−∞, −3) and (−0.7, 1.5).
Specific Function Analysis
- For the function f(x) = (x + 4)²(x − 2)², the y-intercept is at (0, 64) and x-intercepts occur at (-4, 0) and (2, 0).
- Zeros of this function are x = -4 and x = 2, showcasing repeated factors since both occur as even powers.
Translations and Transformations
- Applying a translation affects the function's position on the graph; moving up or down is due to vertical shifts, e.g., f(x) = x⁴ + 2x³ → g(x) = x⁴ + 2x³ + 4 reflects an upward shift of 15 units.
- Horizontal shifting occurs when functions are transformed, e.g., f(x) = x² + 8 to g(x) = (x + 3)² + 8 indicates a left shift by 3 units.
Dilation Effects
- Dilation modifies the steepness or flatness of the graph; a factor less than 1 indicates compression towards the x-axis (e.g., f(x) = 6(x − 3)² + 12 to g(x) = 2(x − 3)² + 4 compresses by 1/3).
- Conversely, factors greater than 1 show stretching away from the x-axis, e.g., g(x) = (9x)³ − 8 stretches f(x) = (3x)³ − 8 by a factor of 1/3.
Mixed Transformations
- When transformations combine, like a reflection and vertical compression, the result is a change in the graph's appearance, e.g., f(x) = −2(x − 1)⁴ + 7 to g(x) = (x − 1)⁴ − 72 involves both a reflection and compression.
Graph Representation
- Understanding how to represent functions graphically is essential. Graphing transformations require recognizing vertical and horizontal shifts and stretching or compressing effects initiated by parameter changes in the function's equation.
General Function Rules
- Given a function's rule, determining a new rule after a transformation is crucial for analyzing changes, such as shifting or dilating, that affect the graph's characteristics and position.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of graphs and functions with these flashcards from Algebra 2A, Unit 4. The quiz covers essential concepts like x-intercepts, y-intercepts, and the domain and range of functions. Prepare for your exam by reviewing these critical topics!
