worksheet b (topic 3.8) the tangent function\n1. a portion of the graph of the function f is shown in the xy…

worksheet b (topic 3.8) the tangent function\n1. a portion of the graph of the function f is shown in the xy - plane, where f is the result of multiplicative transformation of the graph of y = tan x. what is the period of f?\n2. the function g is given by g(x)=2 + 4tan(3x). what is the period of g?\n3. the function m is given by m(θ)= - 2tan(θ/4). what is the period of m?\n4. the function h is given by h(θ)=tan(π/2θ). what is the period of h?\n5. the function k is given by k(x)=tan(x - 2π). what is the period of k?\n6. the function j is given by j(x)=4 + tan(x/2). which of the following gives the vertical asymptotes of j? (a) x = π/2+πk, where k is an integer (b) x = π+2πk, where k is an integer (c) x = π/4+π/2k, where k is an integer (d) x = πk, where k is an integer\n7. the function f is given by f(x)=tan(4x). which of the following gives the vertical asymptotes of f? (a) x = π/8+π/4k, where k is an integer (b) x = π/8+π/2k, where k is an integer (c) x = π/2+πk, where k is an integer (d) x = 2π+4πk, where k is an integer\nworksheet b: topic 3.8\nthe tangent function

worksheet b (topic 3.8) the tangent function\n1. a portion of the graph of the function f is shown in the xy - plane, where f is the result of multiplicative transformation of the graph of y = tan x. what is the period of f?\n2. the function g is given by g(x)=2 + 4tan(3x). what is the period of g?\n3. the function m is given by m(θ)= - 2tan(θ/4). what is the period of m?\n4. the function h is given by h(θ)=tan(π/2θ). what is the period of h?\n5. the function k is given by k(x)=tan(x - 2π). what is the period of k?\n6. the function j is given by j(x)=4 + tan(x/2). which of the following gives the vertical asymptotes of j? (a) x = π/2+πk, where k is an integer (b) x = π+2πk, where k is an integer (c) x = π/4+π/2k, where k is an integer (d) x = πk, where k is an integer\n7. the function f is given by f(x)=tan(4x). which of the following gives the vertical asymptotes of f? (a) x = π/8+π/4k, where k is an integer (b) x = π/8+π/2k, where k is an integer (c) x = π/2+πk, where k is an integer (d) x = 2π+4πk, where k is an integer\nworksheet b: topic 3.8\nthe tangent function

Answer

Explanation:

Step1: Recall period formula for $y = A\tan(Bx - C)+D$

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

Step2: Solve for period of $g(x)=2 + 4\tan(3x)$

Here $B = 3$, so $T=\frac{\pi}{|3|}=\frac{\pi}{3}$.

Step3: Solve for period of $m(\theta)=-2\tan(\frac{\theta}{4})$

Here $B=\frac{1}{4}$, so $T=\frac{\pi}{\left|\frac{1}{4}\right|}=4\pi$.

Step4: Solve for period of $h(\theta)=\tan(\frac{\pi}{2}\theta)$

Here $B = \frac{\pi}{2}$, so $T=\frac{\pi}{\left|\frac{\pi}{2}\right|}=2$.

Step5: Solve for period of $k(x)=\tan(x - 2\pi)$

Here $B = 1$, so $T=\frac{\pi}{|1|}=\pi$.

Step6: Find vertical asymptotes of $y = \tan(u)$

The vertical asymptotes of $y=\tan(u)$ occur at $u=\frac{\pi}{2}+\pi k$, where $k$ is an integer.

Step7: Find vertical asymptotes of $j(x)=4+\tan(\frac{x}{2})$

Set $\frac{x}{2}=\frac{\pi}{2}+\pi k$. Solving for $x$ gives $x=\pi + 2\pi k$, so the answer is B.

Step8: Find vertical asymptotes of $f(x)=\tan(4x)$

Set $4x=\frac{\pi}{2}+\pi k$. Solving for $x$ gives $x=\frac{\pi}{8}+\frac{\pi}{4}k$, so the answer is A.

Answer:

  1. Period of $g(x)$: $\frac{\pi}{3}$
  2. Period of $m(\theta)$: $4\pi$
  3. Period of $h(\theta)$: $2$
  4. Period of $k(x)$: $\pi$
  5. For $j(x)$, vertical - asymptotes: B. $x=\pi + 2\pi k$, where $k$ is an integer
  6. For $f(x)$, vertical - asymptotes: A. $x=\frac{\pi}{8}+\frac{\pi}{4}k$, where $k$ is an integer