Asymptote Architects Worksheet
Asymptote Architects
Structural Analysis: Linear Denominators
NAME: __________________________
DATE: __________________________
Analyze each rational function below. Identify all removable discontinuities (holes), vertical asymptotes (VA), and horizontal asymptotes (HA). Show all work.
1
\(f(x) = \frac{4}{x - 2}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
2
\(f(x) = \frac{2x + 6}{x + 3}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
3
\(f(x) = \frac{3x - 9}{x + 1}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
4
\(f(x) = \frac{x^2 - 16}{x - 4}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
5
\(f(x) = \frac{x + 2}{x - 5}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
6
\(f(x) = \frac{5x}{2x + 10}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
7
\(f(x) = \frac{x^2 - 2x - 3}{x - 3}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
8
\(f(x) = \frac{7}{3x - 6}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
9
\(f(x) = \frac{4x^2 - 4x}{x - 1}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
10
\(f(x) = \frac{-2x + 8}{x - 4}\)
Hole(s):
Vertical Asymptote:
Horizontal Asymptote:
Calculations / Show Work
Architect's Checklist
1. REMOVABLE DISCONTINUITY (HOLE)
Occurs when a factor \((x - c)\) is present in both the numerator and denominator. The graph has a hole at \(x = c\).
2. VERTICAL ASYMPTOTE (VA)
Occurs at any value that makes the simplified denominator equal to zero.
3. HORIZONTAL ASYMPTOTE (HA)
Determined by comparing the degrees of the numerator (\(n\)) and denominator (\(d\)):
- If \(n < d\), HA is \(y = 0\).
- If \(n = d\), HA is \(y = \frac{\text{leading coeff.}}{\text{leading coeff.}}\).
- If \(n > d\), there is no HA.
Asymptote Architects Answer Key
Architect's Master Plan
Answer Key: Linear Denominator Edition
Teacher Resource
1
\(f(x) = \frac{4}{x - 2}\)
Hole(s) None
VA \(x = 2\)
HA \(y = 0\)
2
\(f(x) = \frac{2x + 6}{x + 3}\)
Hole(s) \(x = -3\)
VA None
HA \(y = 2\)
3
\(f(x) = \frac{3x - 9}{x + 1}\)
Hole(s) None
VA \(x = -1\)
HA \(y = 3\)
4
\(f(x) = \frac{x^2 - 16}{x - 4}\)
Hole(s) \(x = 4\)
VA None
HA None
5
\(f(x) = \frac{x + 2}{x - 5}\)
Hole(s) None
VA \(x = 5\)
HA \(y = 1\)
6
\(f(x) = \frac{5x}{2x + 10}\)
Hole(s) None
VA \(x = -5\)
HA \(y = 2.5\)
7
\(f(x) = \frac{x^2 - 2x - 3}{x - 3}\)
Hole(s) \(x = 3\)
VA None
HA None
8
\(f(x) = \frac{7}{3x - 6}\)
Hole(s) None
VA \(x = 2\)
HA \(y = 0\)
9
\(f(x) = \frac{4x^2 - 4x}{x - 1}\)
Hole(s) \(x = 1\)
VA None
HA None
10
\(f(x) = \frac{-2x + 8}{x - 4}\)
Hole(s) \(x = 4\)
VA None
HA \(y = -2\)
Teacher's Quick Notes
HA Rules (Linear Denominators)
- Constant Num / Linear Denom: always \(y = 0\).
- Linear Num / Linear Denom: ratio of coefficients.
- Quadratic Num / Linear Denom: No HA (Slant).
Points to Note
Problems 2, 4, 7, 9, and 10 all feature removable discontinuities. Ensure students specify the x-coordinate of the hole.
