which of the following is the graph of $y = 3\\sec\\left2\\left(x - \\frac{\\pi}{2}\\right)\\right + 2$?
which of the following is the graph of $y = 3\\sec\\left2\\left(x - \\frac{\\pi}{2}\\right)\\right + 2$?
Answer
Explanation:
Step1: Simplify the Function
First, simplify the argument of the secant function: ( y = 3\sec\left[2\left(x - \frac{\pi}{2}\right)\right]+2 = 3\sec(2x - \pi)+2 ). Using the identity ( \sec(\theta - \pi)=-\sec\theta ), we get ( y = - 3\sec(2x)+2 ).
Step2: Analyze the Transformations
- Vertical Stretch: The coefficient ( 3 ) vertically stretches the secant graph by a factor of ( 3 ).
- Reflection: The negative sign reflects the graph over the ( x )-axis.
- Vertical Shift: The ( +2 ) shifts the graph up by ( 2 ) units.
- Period: The period of ( \sec(Bx) ) is ( \frac{2\pi}{|B|} ), so here ( B = 2 ), period is ( \pi ).
Step3: Compare with the Graphs
The graph should have vertical asymptotes at ( 2x = \frac{\pi}{2}+k\pi ) (i.e., ( x=\frac{\pi}{4}+\frac{k\pi}{2} ) for integer ( k )), a vertical shift up by ( 2 ), reflection over ( x )-axis, and period ( \pi ). The first graph (with upper and lower branches) matches these transformations: the upper branches are shifted up, lower branches (after reflection) are present, and the period between asymptotes is ( \pi ).
Answer:
The graph shown in the upper - lower branch structure (the first graph in the provided image, with x - axis marked from (-2\pi) to (2\pi) and y - axis with values like 4, 8, 12, etc. and (-4, - 8, - 12) etc.)