Questions and Answers
What are the graphs of sine functions?
Graphs that represent the sine function.
Which are graphs of sine functions? (Select all that apply)
What is the midline of the function?
y = 2
What is the amplitude of the function?
Signup and view all the answers
What is the amplitude of the function?
Signup and view all the answers
What is the frequency of the function?
Signup and view all the answers
What is the frequency of the function?
Signup and view all the answers
What is the period of the function?
Signup and view all the answers
What is the period of the function?
Signup and view all the answers
What is the rule that represents the graph of the function?
Signup and view all the answers
What is the rule that represents the graph of the function?
Signup and view all the answers
What is the rule that represents the graph of the function?
Signup and view all the answers
Which graph represents the function rule y = 4 cos(2x) - 1?
Signup and view all the answers
Which graphs represent cosine functions? (Select all that apply)
Signup and view all the answers
What is the midline of the function?
Signup and view all the answers
What is the amplitude of the function?
Signup and view all the answers
What is the period of the function?
Signup and view all the answers
What is the frequency of the function?
Signup and view all the answers
Which graph represents the function rule y = 2 cos(3x) + 6?
Signup and view all the answers
Which graphs represent tangent functions? (Select all that apply)
Signup and view all the answers
What is the period of the tangent function?
Signup and view all the answers
Which equations represent vertical asymptotes of the function? (Select all that apply)
Signup and view all the answers
Which equation represents the function in the graph?
Signup and view all the answers
What is the highest average temperature in Phoenix that year (approximately)?
Signup and view all the answers
What is the transformation rule that makes f(x) become g(x)?
Signup and view all the answers
What happens when the function f(x) = sin(x) is transformed by the rule g(x) = f(x) - 4?
Signup and view all the answers
What happens when the function f(x) = cos(x) is transformed by the rule g(x) = f(3x)?
Signup and view all the answers
Study Notes
Graphs of Sine Functions
- The midline of the function can be determined, e.g., y = 2 represents the midpoint of a sine graph.
- Amplitude measures the height from the midline to the peak or trough. Values are represented by specific numbers, e.g., 3, 2, or 4.
- Frequency determines how many cycles fit into a specified interval. Notable frequencies include values like 4 or 2 for various functions.
- The period indicates the length of one complete cycle in a trigonometric function, such as π or 4π.
- Function rules like 2 sin(3x) + 4 and sin(2x) - 3 express transformations in sine graphs.
Graphs of Cosine Functions
- Cosine graphs can be characterized by their midline, amplitude, period, and frequency.
- The amplitude can be found in functions like 3 or 4 cos(2x) + 6 and can also show different transformations.
- Period examples include π/2 and 2π/3, indicating distances between repeating features in these graphs.
- Cosine function rules are seen in expressions such as y = 4 cos(7x) - 1.
Graphs of Tangent Functions
- Tangent functions demonstrate unique properties, including a period of π/2.
- Vertical asymptotes occur at multiples of π/4, specifically noted as x = pi/4, 3pi/4, and so forth.
- Function representations include y = tan(2x) and y = tan(3x), showcasing different transformations and their resulting graphs.
Modeling with Trigonometric Functions
- Certain sine functions may model multiple cycles over given intervals; for example, f(x) = sin(2x) completes 2 cycles over (0, 2π].
- Cosine functions can be tailored to specific attributes, such as f(x) = 2 cos(x) + 1 for a midline or f(x) = 7 cos(x) + 4 for amplitude.
- The height of objects modeled by cosine functions, like a Ferris wheel, can be represented through functions such as f(x) = -106 cos(4x) + 106.
- Cycling frequency in trigonometric functions translates to practical questions like the number of rotations, e.g., a yo-yo represented by f(x) = 2 cos(8πx) + 2 completes 4 rotations in 11 seconds.
Transformations of Trigonometric Functions
- Transformations involve shifting, stretching, or compressing the original functions. For instance, g(x) = f(x + π/4) indicates a leftward phase shift.
- When f(x) = sin(x) is translated down by 4 units, it follows the transformation g(x) = f(x) - 4.
- Various transformations can also involve reflections, such as g(x) = f(−x), which showcases an over-the-y-axis reflection.
- The transformation g(x) = f(2x) demonstrates a horizontal compression by a factor of 2.
Summary of Function Rules
- Specific function rules are highlighted, including transformation rules for adjusted functions like g(x) = -3 cos(2x) + 2, representing reflections and translations in the graphs.
- Study the specific attributes and transformations required for each sine, cosine, and tangent function, as each contributes uniquely to the understanding of trigonometric behavior in different scenarios.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Dive into the graphs of sine functions with this quiz designed for Algebra 2B, Unit 4. Each flashcard includes essential definitions and applications of sine function graphs. Test your understanding and mastery of key concepts related to trigonometric functions.
