12. what is the period of the function given? (lesson 18.2) (0.5 point) a π/2 b π c 2π 13. which of the…

Explanation:

Step1: Recall period - formula for tangent function

The period of the tangent function $y = A\tan(Bx - C)+D$ is $T=\frac{\pi}{|B|}$.

Step2: Determine the period for question 12

For a standard tangent - like function, if we assume the function is of the form $y = \tan(Bx)$, from the graph, we can see that the distance between two consecutive vertical asymptotes gives the period. For the tangent function $y=\tan(Bx)$, the period $T = \frac{\pi}{|B|}$. If we consider the general form of the tangent function and observe the graph, we know that for $y = \tan(Bx)$, when $B = 1$, the period is $\pi$. But if the function is $y=\tan\left(\frac{1}{2}x\right)$, then $T=\frac{\pi}{\left|\frac{1}{2}\right|}=2\pi$.

Step3: Identify the function for question 13

The general form of a tangent function is $y = A\tan(Bx - C)+D$. The vertical shift is given by $D$. The period is $T=\frac{\pi}{|B|}$. The graph has a vertical shift of $D = 2$. The period of the graph is $2\pi$. For the tangent function $y=\tan(Bx)$, since $T=\frac{\pi}{|B|}=2\pi$, then $B=\frac{1}{2}$. So the function is $f(x)=\tan\left(\frac{1}{2}x\right)+2$.

Step4: Find mid - line for question 14a

The general form of a cosine function is $y = A\cos(Bx - C)+D$. The mid - line of the cosine function $y = A\cos(Bx - C)+D$ is given by $y = D$. For the function $f(x)=5\cos\left(x-\frac{\pi}{2}\right)-2$, $D=-2$. So the mid - line is $y=-2$.

Step5: Find amplitude for question 14b

The amplitude of the cosine function $y = A\cos(Bx - C)+D$ is given by $|A|$. For the function $f(x)=5\cos\left(x-\frac{\pi}{2}\right)-2$, $A = 5$, so the amplitude is $|A| = 5$.

Answer:

12. C $2\pi$
13. A $f(x)=\tan\left(\frac{1}{2}x\right)+2$
14. a. $y=-2$ b. $5$