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What does the amplitude of a sine graph tell you?
It indicates the maximum displacement of the graph from its midline, representing the maximum value the function attains.
What does the period of a cosine graph tell you?
It represents the length of one complete cycle of the cosine function before the pattern repeats.
How can you identify the phase shift from a sine or cosine graph?
By observing the horizontal displacement of the graph compared to the standard sine or cosine function. Where does the cycle BEGIN relative to the origin?
How does the graph of $-\sin(x)$ differ from the graph of $\sin(x)$?
The graph of $-\sin(x)$ is a reflection of the graph of $\sin(x)$ over the x-axis.
How does the graph of $-\cos(x)$ differ from the graph of $\cos(x)$?
The graph of $-\cos(x)$ is a reflection of the graph of $\cos(x)$ over the x-axis.
What does a vertical shift in a sine or cosine graph indicate?
It indicates that the entire graph has been moved up or down, changing the midline of the function.
How can you determine the equation of a sine or cosine function from its graph?
Identify the amplitude, period, phase shift, and vertical shift, and then plug these values into the general form of the sine or cosine function.
What does the steepness of a sine or cosine graph indicate?
The steepness relates to how quickly the function's value is changing at that point. Steeper sections indicate more rapid change.
How are the x-intercepts of a sine or cosine graph related to the unit circle?
The x-intercepts correspond to the angles on the unit circle where the sine (y-coordinate) or cosine (x-coordinate) is equal to zero.
How can you tell if a sine or cosine graph has been stretched or compressed horizontally?
By examining the period of the graph. A shorter period indicates horizontal compression, while a longer period indicates horizontal stretching.
What are the key differences between the graphs of $y = \sin(x)$ and $y = \cos(x)$?
$\sin(x)$: Starts at (0,0). | $\cos(x)$: Starts at (0,1).
Compare the symmetry of sine and cosine functions.
Sine: Odd function, symmetric about the origin. | Cosine: Even function, symmetric about the y-axis.
Compare the x-intercepts of $y = \sin(x)$ and $y = \cos(x)$ in the interval $[0, 2\pi]$.
$\sin(x)$: 0, $\pi$, $2\pi$ | $\cos(x)$: $\frac{\pi}{2}$, $\frac{3\pi}{2}$
Compare the maximum values of $y = \sin(x)$ and $y = \cos(x)$.
$\sin(x)$: Maximum value of 1 at $\frac{\pi}{2}$ | $\cos(x)$: Maximum value of 1 at 0 and $2\pi$
Compare the minimum values of $y = \sin(x)$ and $y = \cos(x)$.
$\sin(x)$: Minimum value of -1 at $\frac{3\pi}{2}$ | $\cos(x)$: Minimum value of -1 at $\pi$
Compare the effect of a positive phase shift on $\sin(x)$ and $\cos(x)$.
Both shift the graph to the right by the amount of the phase shift. | The overall shape remains the same, just translated.
Compare the effect of changing the amplitude of $\sin(x)$ and $\cos(x)$.
Both stretch or compress the graph vertically. | A larger amplitude makes the peaks and troughs more extreme.
Compare the effect of changing the period of $\sin(x)$ and $\cos(x)$.
Both compress or stretch the graph horizontally. | A smaller period means more cycles within the same interval.
Compare the effect of a vertical shift on $\sin(x)$ and $\cos(x)$.
Both move the entire graph up or down by the shift amount. | The midline of the graph changes accordingly.
Compare the relationship between sine and cosine to the unit circle.
Sine: y-coordinate on the unit circle. | Cosine: x-coordinate on the unit circle.
What is the general form of a sine function?
$f(x) = A\sin(Bx - C) + D$, where A is amplitude, B affects the period, C is the phase shift, and D is the vertical shift.
What is the general form of a cosine function?
$f(x) = A\cos(Bx - C) + D$, where A is amplitude, B affects the period, C is the phase shift, and D is the vertical shift.
How do you calculate the period of a sine or cosine function given 'B'?
Period = $\frac{2\pi}{|B|}$
How is the phase shift calculated in the general form $f(x) = A\sin(Bx - C) + D$?
Phase Shift = $\frac{C}{B}$
What is the sine of 0?
$\sin(0) = 0$
What is the cosine of 0?
$\cos(0) = 1$
What is the sine of $\frac{\pi}{2}$?
$\sin(\frac{\pi}{2}) = 1$
What is the cosine of $\frac{\pi}{2}$?
$\cos(\frac{\pi}{2}) = 0$
What is the sine of $\pi$?
$\sin(\pi) = 0$
What is the cosine of $\pi$?
$\cos(\pi) = -1$