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How to identify the period from a graph?
Measure the distance for one complete cycle.
How to identify the midline from a graph?
Average of the maximum and minimum y-values.
How to identify the amplitude from a graph?
Half the difference between max and min y-values.
What does a shift in the midline indicate?
A vertical translation of the function.
How does changing the period affect the graph?
It stretches or compresses the graph horizontally.
How does changing the amplitude affect the graph?
It stretches or compresses the graph vertically.
How to identify concave up sections?
Curve looks like a smile.
How to identify concave down sections?
Curve looks like a frown.
What does the graph of \(y = sin(x)\) tell us about its derivative?
The derivative, \(cos(x)\), represents the slope of \(sin(x)\) at each point.
What does the graph of \(y = cos(x)\) tell us about its derivative?
The derivative, \(-sin(x)\), represents the slope of \(cos(x)\) at each point.
Define sinusoidal function.
A function resembling sine or cosine, oscillating and periodic.
What is the period (T) of a sinusoidal function?
The length of one complete cycle.
Define frequency (f).
Number of cycles per unit time; reciprocal of period.
What is the midline (k)?
The horizontal line that cuts the wave in half.
Define amplitude.
Vertical distance from midline to max/min value.
What is odd symmetry?
Symmetry about the origin; f(-x) = -f(x).
What is even symmetry?
Symmetry about the y-axis; f(-x) = f(x).
Define oscillation in sinusoidal functions.
Up and down movement between two values, creating a wave.
What does concave up mean?
The curve looks like a smile.
What does concave down mean?
The curve looks like a frown.
Explain the relationship between sine and cosine.
Cosine is a sine function shifted left by $\frac{ฯ}{2}$.
Explain the concept of period.
The length of one complete cycle before repetition.
Explain the concept of frequency.
How many cycles occur in one unit of time.
Explain how the midline affects the graph.
It shifts the graph vertically up or down.
Explain how amplitude affects the graph.
It stretches or compresses the graph vertically.
Describe the symmetry of sine.
Odd symmetry; symmetric about the origin.
Describe the symmetry of cosine.
Even symmetry; symmetric about the y-axis.
Explain the effect of transformations on the period.
Horizontal stretches/compressions change the period.
Explain the effect of transformations on the amplitude.
Vertical stretches/compressions change the amplitude.
Relationship between period and frequency.
They are reciprocals of each other.