Write an equation of a secant graph that has amplitude of 3, period of pi/2, is shifted to the left pi/4, and is flipped over the x - axis.
To write the equation of a secant graph with the given characteristics, we can start with the general equation of a secant function:
y = Asec(Bx - C) + D,
where A represents the amplitude, B represents the reciprocal of the period, C represents the phase shift, and D represents the vertical shift.
Given:
Amplitude (A) = 3
Period = pi/2
Phase shift (C) = -pi/4
Flipped over x-axis
1. Amplitude: The amplitude represents the vertical distance between the highest and lowest points on the graph. In this case, the amplitude is 3.
So far, the equation becomes:
y = 3sec(Bx - C) + D.
2. Period: The period is the horizontal distance needed for one complete cycle of the graph. The period is the reciprocal of B, so B = 2pi / period.
In this case, the period is pi/2, so we substitute it into the equation to find B:
B = 2pi / (pi/2) = 4.
Thus, the equation becomes:
y = 3sec(4x - C) + D.
3. Phase Shift: The phase shift represents the horizontal displacement of the graph. If the graph is shifted to the left by pi/4 units, we add pi/4 to the argument of the secant function.
So, the equation becomes:
y = 3sec(4x - (-pi/4)) + D
= 3sec(4x + pi/4) + D.
4. Flipped over x-axis: To flip the graph over the x-axis, we multiply the entire equation by -1.
Therefore, the final equation becomes:
y = -3sec(4x + pi/4) + D.
Note: The vertical shift (D) is not mentioned in the given characteristics. If the vertical shift is not specified, it is assumed to be zero (0).