The amplitude determines the vertical stretch of the graph. A larger 'a' means a greater distance between the max/min and the midline.

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.

The vertical shift moves the entire graph up or down. The midline is shifted to y = d.

The horizontal shift moves the graph left or right. A positive 'c' shifts the graph left, and a negative 'c' shifts the graph right.

If the graph starts at the midline going up, use sine. If it starts at a maximum, use cosine.

The value of 'b' is inversely proportional to the period. A larger 'b' results in a shorter period, and vice versa.

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 $f(x) = a\cos(b(x + c)) + d$.

It reflects the graph over the midline.

Changing 'b' compresses or stretches the graph horizontally, affecting the period.

'c' shifts the graph horizontally; a positive 'c' shifts left, and a negative 'c' shifts right.

$f(x) = a \sin(b(x + c)) + d$

$f(x) = a \cos(b(x + c)) + d$

$amplitude = \frac{|max - min|}{2}$

$b = \frac{2\pi}{period}$

$d = \frac{max + min}{2}$

Period = $\frac{2\pi}{b}$, where b is frequency.

$midline = \frac{max + min}{2}$

$b = \frac{2\pi}{Period}$

$Period = \frac{2\pi}{b}$

$max = d + |a|$

$min = d - |a|$

1. Identify the max and min y-values. 2. Use the formula: $amplitude = \frac{|max - min|}{2}$.

1. Identify one complete cycle. 2. Measure the horizontal distance of that cycle.

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'.

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.

1. Find the period. 2. Use the formula $b = \frac{2\pi}{period}$.

1. Find a, b, c, and d. 2. Use the general form: $f(x) = a\sin(b(x + c)) + d$.

Maximum: d + |a|. Minimum: d - |a|.

1. Substitute x, y, a, b, and d into $f(x) = a\sin(b(x + c)) + d$. 2. Solve for 'c'.

If 'a' is negative, the function is reflected over the x-axis.

1. Calculate amplitude and vertical shift. 2. Calculate 'b' from the period. 3. Determine the phase shift (if any). 4. Write the equation.