$\oint 8.1$ Direct, Indirect, and Joint Variation

Direct Variation: Both UP or both DOWN

$y$ varies directly with $x$, means $y = kx$, where $k$ is called a constant of variation

Example: surface area of a cube varies directly with the edge length of the cube $$A = 6(l^2)$$

Indirect Variation: One UP and one DOWN

$y$ varies inversely as $x$, means $\displaystyle y = \frac{k}{x}$

General Solution vs. Specific Solution(Example)

$x$ and $y$ vary inversely

$y = 7$

$x = 4$

General Solution Specific Solution
$\displaystyle y = \frac{k}{x}$ $\displaystyle y = \frac{28}{x}$

the general solution is the equation with $k$ still in it the specific solution is solved for $k$

Joint Variation indicates that multiple variables are related to each other

$y$ varies directly with $x$ and inversely with $z$

$\displaystyle y = \frac{kx}{z}$

if $y = -6, x = 2, $ and $ z = -1$

then $\displaystyle y = \frac{3x}{z}$

Example

$p$ varies directly with the cube of $w$ and the square root $z$, and inversely with the fifth power of $q$

$$p = \frac{kw^3\sqrt{z}}{q^5}$$

$\oint 8.2$ Graphing Simple Rational Functions

A rational function with polynomials $p(x)$ & $q(x)$ with $q(x) \neq 0$ will be in the form

$$f(x) = \frac{p(x)}{q(x)}$$
$$f(x) = \frac{a}{x-h}+k$$

$h$ = horizontal shift and asymptote

$k$ = vertical shift and asymptote

Example:

$\displaystyle f(x) = \frac{1}{x}$

Vertical asymptote at $x=0$

Horizontal asymptote at $y=0$

As $x \to \infty \mid f(x) \approx 0$

As $x \to -\infty \mid f(x) \approx 0$

Example 2:

$\displaystyle y = \frac{1}{x-3}+2$

Vertical asymptote at $x=2$

Horizontal asymptote at $y=3$

Domain: $x\in \mathbb R \ni x \neq 3$

Range: $y\in \mathbb R \ni y \neq 2$

Sample Graph

$$ y = \frac{1}{x-3} + 2 $$
In [27]:
%matplotlib inline

import matplotlib
import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-5, 5, 500)
axes = plt.gca()
axes.set_xlim([-5,5])
axes.set_ylim([-5,5])
plt.plot(x, (1 / (x-3) ) + 2)
plt.grid(True)
plt.title('y = 1/(x-3) + 2')
plt.show()

$\oint 8.3$ Graphing Rational Functions

Two types of discontinuity

  1. Non-removal
    • $f(x) = \frac{1}{x} \to x = 0$ is vertical asymptote
  2. Removable Discontinuity
    • "Point of Removable Discontinuity" AKA Hola
    • $f(x) = \frac{(x+1)(x-3)}{(x+1)}$
    • $x = -1 \to (-1,-4)$ is a hole
$$ f(x) = \frac{p(x)}{q(x)} $$
  1. Degree $p(x) > q(x)$
    • No Horizontal Asymptote
  2. Degree $p(x) < q(x)$
    • Horizontal Asymptote at $y=0$
  3. Degree $p(X) = q(x)$
    • Horizontal Asymptote at $y=$ leading coefficient
    • $f(x) = \frac{3x+5}{7-8x} \to HA @ y = -\frac{3}{8}$

$\oint 8.4$ Rational Expressions

To Simplify:

  1. Factor both numerator and denominator
  2. Cancel common factors

$\oint 8.5$ Add and Subtract Rational Expressions

  1. Find the LCD

$\oint 8.6$ Solving Rational Equations

  1. Determine domain restrictions
  2. Solve
  3. Check Solution

Steps for Solving Algebraically:

  1. Rewrite the inequality as a single expression compared to zero
  2. Identify the critical x-values
  3. Test an x-value in each region on the number line to determine its sign $(+/-)$
  4. Write the inequalities that describe the solution
In [ ]: