Questions and Answers
The graph of which function passes through (0,4) and has a minimum value at (3π/2, 3)?
f(x) = sin(x) + 4
Which function describes the graph shown below?
y = 1/2sin(x) - 2
Which function describes the graph below?
y = 4cos(x) + 3
Which function is graphed below?
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What is the period of f(x) = sin(x)?
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Which function has the given properties? The domain is the set of all real numbers. One x-intercept is given, the amplitude is 4, the point is on the graph, the y-intercept is (0, 0).
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The range of the function f(x) = 5cos(x) + 1 is the set of real numbers -4 ≤ y ≤ 6.
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What is the maximum of f(x) = sin(x)?
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Which set of transformations is needed to graph f(x) = -2sin(x) + 3 from the parent sine function?
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What is the range of y = -5sin(x)?
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Study Notes
Sine and Cosine Functions Overview
- The sine function, f(x) = sin(x), has a period of 2π.
- The cosine function, f(x) = cos(x), serves as the basis for cosine graphs.
Graph Characteristics
- The graph f(x) = sin(x) + 4 passes through (0, 4) and has a minimum at (3π/2, 3).
- f(x) = -4sin(x) has a domain of all real numbers with an amplitude of 4 and y-intercept at (0, 0).
Transformation Properties
- The transformation resulting in f(x) = -2sin(x) + 3 involves reflecting across the x-axis, stretching vertically by a factor of 2, and translating vertically by 3 units.
- The function y = 1/2sin(x) - 2 represents a vertical compression and downward translation.
Range and Maximum Values
- The range of f(x) = 5cos(x) + 1 spans from -4 to 6, indicating the highest and lowest values of the function.
- The maximum value of f(x) = sin(x) is 1, illustrating the peak of the sine wave.
- The range of y = -5sin(x) is from -5 to 5, showing the effects of amplitude and reflection.
Specific Function Graphs
- y = 4cos(x) + 3 features a vertical shift upward by 3 units from the basic cosine function.
X-intercepts and Important Points
- The graph of f(x) = sin(x) can illustrate various key points, including the x-intercept and other transformations required for graphing.
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Description
This quiz covers key concepts related to sine and cosine functions, including their graphs, transformation properties, and range of values. You'll explore the periodic nature of these functions and their graph characteristics, helping to deepen your understanding of trigonometry.
