🎯⭐ INTERACTIVE LESSON

Rational Functions and Asymptotes

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Rational Functions and Asymptotes - Complete Interactive Lesson

Part 1: Domain Restrictions

📐 Domain Restrictions

Part 1 of 7 — Domain Restrictions

A rational function is $f(x) = \frac{p(x)}{q(x)}$ where $q(x) \neq 0$ .

Domain: all real numbers except where the denominator = 0.

Set $q(x) = 0$ and solve to find restrictions.

Worked Example

$f(x) = \frac{x+1}{x-3}$ . Domain?

$x - 3 = 0 \Rightarrow x = 3$

Concept Check 🎯

Domain Restrictions 🧮

Find where denominator = 0:

$\frac{1}{x-5}$ . Restricted at $x =$ ?
$\frac{x}{x+2}$ . Restricted at ?

Concept Check 🔍

Practice

#	Function	Restriction
1	$\frac{1}{x-5}$	x ≠ 5
2	$\frac{x}{x+2}$

Challenge Question 📋

Part 2: Vertical Asymptotes

📊 Vertical Asymptotes

Part 2 of 7 — Vertical Asymptotes

A vertical asymptote occurs at $x = a$ when:

The denominator equals zero at $x = a$
The factor does NOT cancel with the numerator

The graph approaches $\pm\infty$ near a vertical asymptote.

Worked Example

. Vertical asymptote?

Part 3: Horizontal Asymptotes

🔢 Horizontal Asymptotes

Part 3 of 7 — Horizontal Asymptotes

Compare degrees of numerator ( $n$ ) and denominator ( $d$ ):

Condition	HA
$n < d$	$y = 0$

Part 4: Holes in Graphs

📈 Holes in Graphs

Part 4 of 7 — Holes in Graphs

A hole occurs when a factor cancels from both numerator and denominator.

$f(x) = \frac{(x-2)(x+1)}{(x-2)(x+3)}$

Part 5: Graphing Rational Functions

🧮 Graphing Rational Functions

Part 5 of 7 — Graphing Rational Functions

Steps:

Find domain restrictions (den = 0)
Identify holes (cancel common factors)
Find VAs (remaining den zeros)
Find HA (compare degrees)
Find x-intercepts (num = 0) and y-intercept ( $f(0)$ )
Plot and connect

Worked Example

$f(x) = \frac{x+2}{x-1}$

Part 6: Problem-Solving Workshop

🛠️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Combine all concepts for rational functions:

Domain, VAs, HAs, holes
Intercepts
Sketch the graph

Worked Example

$f(x) = \frac{2x}{x^2-1} = \frac{2x}{(x-1)(x+1)}$

Part 7: Review & Applications

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Concepts

Domain: exclude den = 0
VA: non-canceled den zeros
HA: compare degrees ( $n<d$ : y=0; $n=d$ : LC ratio; $n>d$ : none)
Holes: canceled common factors
x-int: num = 0; y-int: f(0)

Domain: all reals except $x = 3$ ✅

$\frac{1}{x^2-4}$ . One restriction at $x =$ ?

f(x) = \frac{2}{x-4}

f (x) = \frac{2}{x - 4}

$x - 4 = 0 \Rightarrow x = 4$

Vertical asymptote at $x = 4$ ✅

Concept Check 🎯

Vertical Asymptotes 🧮

$\frac{1}{x-3}$ . VA at $x =$ ?
$\frac{x}{x+1}$ . VA at $x =$ ?
$\frac{1}{(x-2)(x+5)}$ . One VA at $x =$ ?

Concept Check 🔍

Practice

#	Function	VA
1	$\frac{1}{x-3}$	x = 3
2	$\frac{x}{x+1}$	x = −1
3	$\frac{1}{(x-2)(x+5)}$	x = 2, x = −5

Challenge Question 📋

Worked Example

$f(x) = \frac{3x^2}{x^2 + 1}$ . HA?

Degrees equal (both 2). HA: $y = \frac{3}{1} = 3$ ✅

Concept Check 🎯

Horizontal Asymptotes 🧮

$\frac{1}{x+1}$ . HA: $y =$ ?
$\frac{2x}{x-3}$ . HA: $y =$ ?
$\frac{4x^2}{2x^2+1}$ . HA: $y =$ ?

Concept Check 🔍

Practice

#	Function	HA
1	$\frac{1}{x+1}$	y = 0
2	$\frac{2x}{x-3}$	y = 2
3	$\frac{4x^2}{2x^2+1}$	y = 2

Challenge Question 📋

The $(x-2)$ cancels → hole at $x = 2$ .

To find the y-value of the hole, substitute into the simplified function.

Worked Example

$f(x) = \frac{(x-2)(x+1)}{(x-2)(x+3)}$ . Hole?

Cancel $(x-2)$ : simplified = $\frac{x+1}{x+3}$

Hole at $x=2$ : $y = \frac{2+1}{2+3} = \frac{3}{5} = 0.6$ ✅

Concept Check 🎯

Holes 🧮

$\frac{(x-3)(x+1)}{(x-3)(x-1)}$ . Hole at $x =$ ?
$\frac{x(x-5)}{x(x+2)}$ . Hole at $x =$ ?
$\frac{(x-2)(x+1)}{(x-2)(x+3)}$ . y-value of hole?

Concept Check 🔍

Practice

#	Function	Hole at
1	$\frac{(x-3)(x+1)}{(x-3)(x-1)}$	x = 3
2	$\frac{x(x-5)}{x(x+2)}$	x = 0

Challenge Question 📋

No common factors → no holes
VA: $x = 1$
HA: $y = 1$ (equal degrees, 1/1)
x-int: $x = -2$
y-int: $f(0) = \frac{2}{-1} = -2$ ✅

Concept Check 🎯

Graph $\frac{x}{x-2}$ 🧮

VA at $x =$ ?
HA at $y =$ ?
x-intercept at $x =$ ?

Concept Check 🔍

Practice

#	Function	VA	HA	x-int
1	$\frac{x}{x-2}$	2	1	0
2	$\frac{3}{x+1}$	−1	0	none

Challenge Question 📋

VAs: $x = 1, x = -1$
HA: $y = 0$ (degree 1 < degree 2)
x-int: $x = 0$ , y-int: $f(0) = 0$ ✅

Concept Check 🎯

Analyze $\frac{x-3}{x+2}$ 🧮

VA at $x =$ ?
HA at $y =$ ?
x-intercept at $x =$ ?

Concept Check 🔍

Practice

#	Function	Analysis
1	$\frac{x-3}{x+2}$	VA: −2, HA: 1
2	$\frac{4}{x^2}$	VA: 0, HA: 0

Challenge Question 📋

Worked Example

$\frac{3x+6}{x+2} = \frac{3(x+2)}{x+2}$

Common factor → hole at $x=-2$ , simplified: $y = 3$ (Horizontal line with a hole) ✅

Concept Check 🎯

Review 🧮

VA of $\frac{5}{x-4}$ at $x =$ ?
HA of $\frac{2x}{3x+1}$ at $y =$ ? (round to hundredths)
Hole of $\frac{x(x-1)}{x(x+3)}$ at $x =$ ?

Concept Check 🔍

Practice

#	Topic	Problem
1	VA	$\frac{5}{x-4}$
2	HA	$\frac{2x}{3x+1}$
3	Hole	$\frac{x(x-1)}{x(x+3)}$

Challenge Question 📋