Deep Dive into the Graph of 1/(x+1) + 2
Soviet Joke: In Soviet Russia, function graphs you! 🇷🇺
Why x = -1 is the Vertical Asymptote:
- Identify the Denominator: In the function 1/(x+1) + 2, the denominator is x+1.
- Set the Denominator Equal to Zero: To find where the function is undefined, set the denominator x+1 equal to zero.
- Solve for x: Solving the equation x+1 = 0 gives x = -1.
- Vertical Asymptote: Since the function becomes undefined at x = -1, this is where the vertical asymptote occurs.
Additional Graph Features:
- Horizontal Asymptote: The graph has a horizontal asymptote at y = 2. As x moves towards positive or negative infinity, the graph approaches this line but never actually reaches it.
- Quadrants: The graph is located in the 1st and 4th quadrants. This is because the function is shifted two units up.
- End Behavior: As x approaches -1 from the left, the function value (y) dives towards negative infinity. As x approaches -1 from the right, the function value (y) soars towards positive infinity.
- Curve Shape: The graph is hyperbolic, meaning it has a curve that gets closer and closer to the asymptotes but never actually touches them.
- Shift: Compared to the graph of 1/x, this graph is shifted one unit to the left and two units up, due to the “+1” in the denominator and the “+2” added to the function.