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Define amplitude in a sinusoidal function.
Half the distance between the maximum and minimum y-values. Calculated as $\frac{|max - min|}{2}$.
Define the period of a sinusoidal function.
The horizontal distance required for the function to complete one full cycle.
Define the frequency of a sinusoidal function.
Related to the period by $b = \frac{2\pi}{period}$, where 'b' is a parameter in the sinusoidal equation.
Define the vertical shift (d) in a sinusoidal function.
The vertical translation of the midline of the function. It's the 'd' value in the equation $f(x) = a\sin(b(x + c)) + d$.
Define horizontal shift (c) or phase shift in a sinusoidal function.
The horizontal translation of the function, indicating how much the graph is shifted left or right.
What is the midline of a sinusoidal function?
The horizontal line that runs midway between the maximum and minimum values of the function. Calculated as $\frac{max + min}{2}$.
What does 'a' represent in $f(x) = a \sin(b(x + c)) + d$?
Amplitude.
What does 'b' represent in $f(x) = a \sin(b(x + c)) + d$?
Frequency, related to the period.
What does 'c' represent in $f(x) = a \sin(b(x + c)) + d$?
Horizontal shift (phase shift).
What does 'd' represent in $f(x) = a \sin(b(x + c)) + d$?
Vertical shift.
How to find the amplitude from a graph?
1. Identify the max and min y-values. 2. Use the formula: $amplitude = \frac{|max - min|}{2}$.
How to find the period from a graph?
1. Identify one complete cycle. 2. Measure the horizontal distance of that cycle.
How to find the vertical shift from a graph?
1. Identify the max and min y-values. 2. Calculate the midline: $midline = \frac{max + min}{2}$. 3. The midline value is the vertical shift 'd'.
How to find the horizontal shift from a graph?
1. Choose a key point (e.g., starting point of a sine or cosine wave). 2. Compare its position to the standard sine or cosine graph. 3. Determine the horizontal distance it has shifted.
How to determine 'b' from a graph?
1. Find the period. 2. Use the formula $b = \frac{2\pi}{period}$.
How to write the equation of a sine function from a graph?
1. Find a, b, c, and d. 2. Use the general form: $f(x) = a\sin(b(x + c)) + d$.
How to find the maximum and minimum values of $f(x) = a \sin(b(x + c)) + d$?
Maximum: d + |a|. Minimum: d - |a|.
How to find the phase shift 'c' given a point (x, y) on the graph and the values of a, b, and d?
1. Substitute x, y, a, b, and d into $f(x) = a\sin(b(x + c)) + d$. 2. Solve for 'c'.
How to determine if a sinusoidal function is reflected over the x-axis?
If 'a' is negative, the function is reflected over the x-axis.
How to determine the equation of a sinusoidal function given its maximum and minimum values and period?
1. Calculate amplitude and vertical shift. 2. Calculate 'b' from the period. 3. Determine the phase shift (if any). 4. Write the equation.
Explain how the amplitude 'a' affects the graph of a sinusoidal function.
The amplitude determines the vertical stretch of the graph. A larger 'a' means a greater distance between the max/min and the midline.
Explain how the period affects the graph of a sinusoidal function.
The period determines the horizontal length of one complete cycle. A shorter period means the cycles are compressed, and a longer period means they are stretched.
Explain how the vertical shift 'd' affects the graph of a sinusoidal function.
The vertical shift moves the entire graph up or down. The midline is shifted to y = d.
Explain how the horizontal shift 'c' affects the graph of a sinusoidal function.
The horizontal shift moves the graph left or right. A positive 'c' shifts the graph left, and a negative 'c' shifts the graph right.
How do you determine if a sinusoidal function should be modeled with sine or cosine?
If the graph starts at the midline going up, use sine. If it starts at a maximum, use cosine.
What is the relationship between 'b' and the period of a sinusoidal function?
The value of 'b' is inversely proportional to the period. A larger 'b' results in a shorter period, and vice versa.
Explain how to find the equation of a sinusoidal function from its graph.
Determine the amplitude, period, vertical shift, and horizontal shift from the graph. Use these values to construct the equation in the form $f(x) = a\sin(b(x + c)) + d$ or $f(x) = a\cos(b(x + c)) + d$.
How does a negative 'a' value affect the graph?
It reflects the graph over the midline.
What is the impact of changing 'b' on the graph?
Changing 'b' compresses or stretches the graph horizontally, affecting the period.
Describe the impact of 'c' on the graph.
'c' shifts the graph horizontally; a positive 'c' shifts left, and a negative 'c' shifts right.