Homework 4

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. If $\sin t > 0$ and $\cos t < 0$, what quadrant is the terminal point in?
  2. If $\cos t < 0$ and $\cot t < 0$, what quadrant is the terminal point in?
  3. Is the function $f(t) = \sin (t) \cos (t)$ negative or positive when $t$ is in Quadrant IV? How about Quadrant II?
  4. Is the function $f(t) = \frac{\tan t}{\cos t}$ negative or positive when $t$ is in Quadrant III? How about Quadrant I?
  5. Fill out this table. It will probably be helpful in the future.
    $t$ $\tan(t)$
    $0$
    $\frac{\pi}{6}$
    $\frac{\pi}{4}$
    $\frac{\pi}{3}$
    $\frac{\pi}{2}$
  6. $P\left(-\dfrac{3}{5}, -\dfrac{4}{5}\right)$ is a terminal point associated with some $t \in \mathbb{R}$. Find $\sin(t), \cos(t)$ and $\tan(t)$.
  7. Let's say $\cos(t) = -\frac{7}{25}$ and the terminal point of $t$ is in Quadrant III. What is $\sin(t)$ and $\tan(t)$?
  8. Suppose $f(x) = \sqrt{x}$. We want to vertically shift $f(x)$ upwards 3 units and left four units to make a new function $g(x)$. What is the formula of $g(x)$?
  9. Starting from a base function $f(x)$, I want to create a function where I
    1. vertically shift up three units
    2. horizontally shift right four units
    3. vertical stretch of 2 units
    4. horizontal shrink of $\frac{1}{3}$ units
    5. reflection over the $x$-axis
    What should the formula of $f(x)$ look like?
  10. Suppose \[f(x) = \sqrt{x} \qquad g(x) = \sqrt{x - 2} \qquad h(x) = \sqrt{\frac{1}{2}x - 2} \qquad i(x) = \sqrt{\frac{1}{2}\left(x-2\right)}\]
    1. Is $g(x)$ horizontally shifted two units to the right from $f(x)$? Why or why not?
    2. Is $h(x)$ horizontally shifted two units to the right from $f(x)$? Why or why not?
    3. Is $i(x)$ horizontally shifted two units to the right from $f(x)$? Why or why not?
  11. Suppose \[f(x) = x^3 \qquad g(x) = \left(\pi\left(x + 4\right)\right)^3 \qquad h(x) = \left(0.25x + 4\right)^3 \qquad i(x) =(x+4)^3\]
    1. Is $g(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?
    2. Is $h(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?
    3. Is $i(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?
  12. State each relevant transformation, amplitude, and period (amplitude only for $\sin, \cos$). Graph one complete period by hand. Be sure to show in the graph the endpoints of one period.
    1. $y = 1 + \cos (x - \pi)$
    2. $y = \dfrac{1}{2} - \dfrac{1}{2}\cos\left(2x - \dfrac{\pi}{3}\right)$
    3. $y = \dfrac{1}{2} - \dfrac{1}{2}\cos\left(2\left(x - \dfrac{\pi}{3}\right)\right)$
    4. $y = \sin(\pi + 2x)$
    5. (skip this) $y = 1 + \tan\left(\pi x\right)$
    6. (skip this) $y = 4 \tan (3x - 2\pi)$
    7. $y = 5\sin\left(\pi x + \dfrac{\pi}{3}\right)$
  13. Suppose \[f(x) = x \qquad\qquad g(x) = \sin(x)\] Graph $f(x) + g(x)$ in Desmos. Based on what you know about graphical addition, why does the graph $f(x) + g(x)$ look like the way it does?