3.5 - Rational Functions

The relationship between velocity, time, and distance:
- Velocity (v) is defined as the distance traveled over time.
- Common unit: miles per hour (mph).
- The formula relating velocity, distance (d), and time (t) is:
- $d = v \times t$
  (d) represents the total distance traveled.
Example of distance traveled:
- If traveling at 50 miles per hour:
- In the first hour, distance = 50 miles.
- In the second hour, distance = 50 miles.
- Total distance in 2 hours = 100 miles.

To determine the time (t) for a fixed distance (D) given velocity:
- Rearranging the distance formula:
- $t = \frac{D}{v}$
- This expresses time as inversely proportional to velocity.
Inverse relationship:
- As velocity increases (faster speeds), time decreases.
- This exemplifies the concept of inverse proportionality in functions.

Definition of Inversely Proportional Function:
- Example function: $f(x) = \frac{1}{x}$
- As one variable increases, the other decreases; this is a reciprocal function.

Gravitational Force:
- Acceleration due to gravity is approximately $9.8 ext{ m/s}^2$ .
- The relationship is inversely proportional to the square of the distance.
Volume of Sound:
- The volume of sound (v) at a distance (d) can be described by:
- Relationship: $v = \frac{k}{d^2}$
- Here, k is a constant that denotes the medium through which sound travels.
- Engineers account for k when designing soundproofing around microphones.

Graph of Reciprocal Functions:
- General shape:
- $f(x) = \frac{1}{x}$ creates an asymptotic curve near zero.
- Reciprocal squared function: $f(x) = \frac{1}{x^2}$ .
- This will produce curves that do not touch the x-axis due to squaring values.

Short Run Behavior:
- Examines how function graphs behave at specific input values, particularly around discontinuities.
- Vertical Asymptote (VA): occurs at points where f(x) approaches infinity as x approaches zero from either side.
- For $f(x) = \frac{1}{x}$ :
- As x approaches zero, f(x) goes to either positive or negative infinity depending on the direction from which x approaches.
Long Run Behavior:
- Explores function behavior as input (x) approaches infinity.
- Horizontal Asymptote (HA): describes the behavior of f(x) as x approaches positive or negative infinity.
- For $f(x) = \frac{1}{x}$ :
  - As x approaches positive or negative infinity, f(x) approaches 0.

Vertical Asymptote (VA):
- Defined as the vertical line $x=a$ where f(x) approaches positive or negative infinity as x approaches a.
Horizontal Asymptote (HA):
- Defined as the horizontal line $y=b$ where f(x) approaches b as x approaches infinity.

When shifting the graph of $f(x)$ :
- Shifting to the left by 2 units and up by 3 units modifies function:
- $f(x) = \frac{1}{x+2} + 3$
- Requires identifying new asymptotes, which are altered based on transformations.

General form:
- A rational function is represented as:
  $f(x) = \frac{P(x)}{Q(x)}$
- Where P(x) and Q(x) are polynomials.
Key aspect of rational functions is the analysis of asymptotes and intercepts.

VA occurs where the denominator $Q(x) = 0$ while \(P(x)) is non-zero.
- Example: (k(x) = \frac{5 + 2x^2}{2 - x - x^2} $</p></li><li><p>Set the denominator$ Q(x) = 0 to find VAs
- Solving provides values where the function is undefined.

Long run behavior is determined by comparing the degrees of the numerator and denominator:
- Case 1: Degree of $P(x)$ < Degree of $Q(x)$: HA at y=0.
- Case 2: Degree of $P(x)$ = Degree of $Q(x)$: HA at \frac{leading coefficient of P}{leading coefficient of Q}$$.
- Case 3: Degree of $P(x)$ > Degree of $Q(x)$: No horizontal asymptote; check for oblique asymptotes instead.

Understanding the relationship between functions and their asymptotic behavior is crucial in analyzing rational functions across various contexts, from basic physics to advanced calculus.
The focus on behavior near vertical and horizontal asymptotes aids in graphing and conceptualizing rational function dynamics.