Asymptotes & Rational Functions
Explore boundary singularities and hyperbola branches. Shift coordinate asymptotes, launch particles into the singularity field, and calculate numerical limits.
Magnet Barriers
Deflection Field Sandbox
A rational function of form $y = a/(x-h) + k$ has two asymptotic "walls" where values approach infinity. Drag the blue pegs on the axes directly to shift the asymptotes, or adjust the parameters using sliders.
Drag the blue square pegs on the X and Y axes to alter horizontal/vertical asymptotes.
Active Rational Equation
y = 1.00 / (x + 2.0) + 2.0
Active Limit Values:
x → ±∞, y → 2.00
x → -2.00⁺, y → +∞
x → -2.00⁻, y → -∞
x → -2.00⁺, y → +∞
x → -2.00⁻, y → -∞
Anatomy Specs
Asymptotes, Domain & Range
A rational function exhibits singular discontinuities where the denominator approaches zero, leading to division by zero.
Symmetry and Breaks
Vertical Asymptote:
Occurs at **x = -2.00**. The denominator becomes zero, rendering the function completely **undefined** at this coordinate.
Horizontal Asymptote:
Occurs at **y = 2.00**. As x increases towards infinity, the fractional term values diminish towards zero, leaving y near the vertical shift parameter.
Domain & Range bounds:
Domain: All real numbers **x ≠ -2.00**
Range: All real numbers **y ≠ 2.00**
Range: All real numbers **y ≠ 2.00**
Intercept Calculations
Y-Intercept (x = 0):
y = a/(-h) + k = 1.00 / (2.0) + 2.0 = **2.50**
Coordinate: **(0, 2.50)**
Coordinate: **(0, 2.50)**
X-Intercept (y = 0):
0 = a/(x-h) + k → x = h - a/k = -2 - 1/2 = **-2.50**
Coordinate: **(-2.50, 0)**
Coordinate: **(-2.50, 0)**
Self Evaluation
Rational Functions Quiz
Review your core understanding of limits, asymptotes, and coordinate shifts.
Question 1 of 5
Score: 0/0
What values are excluded from the domain of y = 3 / (x + 4) - 2?
Correct Answer!
Explanation goes here.
Focus Area Vocabulary
- Asymptote: A boundary line that a curve approaches infinitely but never reaches.
- Singularity: Point where denominator becomes zero and function shoots to infinity.
- Hyperbola Branches: The two disconnected symmetrical curved branches forming the graph.