Ultimate guide to sqrt(x), sqrt(x+1), sqrt(x-1) with graphs and detailed information

Unlock the Mysteries of the Square Root Graph: √x

Soviet Joke: In Soviet Russia, square root takes you! 🇷🇺

Identify the Radicand: In the function √x, the radicand (the number under the square root) is x.
Set the Radicand ≥ 0: The square root of a negative number is not defined in the real number system, so x must be greater than or equal to zero.
Domain: The domain of √x is x ≥ 0.

Range: The range of √x is y ≥ 0. The graph starts at the origin (0,0) and extends infinitely upwards and to the right.
Quadrants: The graph is located in the 1st quadrant.
Curve Shape: The graph is a curve that starts at the origin and extends infinitely to the right, getting gradually flatter as x increases.
End Behavior: As x approaches infinity, y also approaches infinity, but at a decreasing rate.

Soviet Joke: In Soviet Russia, square root shifts you! 🇷🇺

Identify the Radicand: In the function √(x-1), the radicand (the number under the square root) is x-1.
Set the Radicand ≥ 0: The square root of a negative number is not defined in the real number system, so x-1 must be greater than or equal to zero.
Solve for x: Solving the inequality x-1 ≥ 0 gives x ≥ 1.
Domain: The domain of √(x-1) is x ≥ 1.

Range: The range of √(x-1) is y ≥ 0. The graph starts at the point (1,0) and extends infinitely upwards and to the right.
Quadrants: The graph is located in the 1st quadrant.
Curve Shape: The graph is a curve that starts at the point (1,0) and extends infinitely to the right, getting gradually flatter as x increases.
End Behavior: As x approaches infinity, y also approaches infinity, but at a decreasing rate.
Shift: Compared to the graph of √x, this graph is shifted one unit to the right due to the “-1” in the radicand.

Soviet Joke: In Soviet Russia, square root shifts you! 🇷🇺

Identify the Radicand: In the function √(x+1), the radicand (the number under the square root) is x+1.
Set the Radicand ≥ 0: The square root of a negative number is not defined in the real number system, so x+1 must be greater than or equal to zero.
Solve for x: Solving the inequality x+1 ≥ 0 gives x ≥ -1.
Domain: The domain of √(x+1) is x ≥ -1.

Range: The range of √(x+1) is y ≥ 0. The graph starts at the point (-1,0) and extends infinitely upwards and to the right.
Quadrants: The graph is located in the 1st and 2nd quadrants.
Curve Shape: The graph is a curve that starts at the point (-1,0) and extends infinitely to the right, getting gradually flatter as x increases.
End Behavior: As x approaches infinity, y also approaches infinity, but at a decreasing rate.
Shift: Compared to the graph of √x, this graph is shifted one unit to the left due to the “+1” in the radicand.

Soviet Joke: In Soviet Russia, square root shifts you! 🇷🇺

Identify the Radicand: In the function √(x+1) – 1, the radicand (the number under the square root) is x+1.
Set the Radicand ≥ 0: The square root of a negative number is not defined in the real number system, so x+1 must be greater than or equal to zero.
Solve for x: Solving the inequality x+1 ≥ 0 gives x ≥ -1.
Domain: The domain of √(x+1) – 1 is x ≥ -1.

Range: The range of √(x+1) – 1 is y ≥ -1. The graph starts at the point (-1,-1) and extends infinitely upwards and to the right.
Curve Shape: The graph is a curve that starts at the point (-1,-1) and extends infinitely to the right, getting gradually flatter as x increases.
End Behavior: As x approaches infinity, y also approaches infinity.
Shift: Compared to the graph of √x, this graph is shifted one unit to the left and one unit down due to the “+1” in the radicand and the “-1” subtracted from the function.

Soviet Joke: In Soviet Russia, square root shifts you! 🇷🇺

Identify the Radicand: In the function √(x-1) + 1, the radicand (the number under the square root) is x-1.
Set the Radicand ≥ 0: The square root of a negative number is not defined in the real number system, so x-1 must be greater than or equal to zero.
Solve for x: Solving the inequality x-1 ≥ 0 gives x ≥ 1.
Domain: The domain of √(x-1) + 1 is x ≥ 1.

Range: The range of √(x-1) + 1 is y ≥ 1. The graph starts at the point (1,1) and extends infinitely upwards and to the right.
Curve Shape: The graph is a curve that starts at the point (1,1) and extends infinitely to the right, getting gradually flatter as x increases.
End Behavior: As x approaches infinity, y also approaches 1, but at a decreasing rate.
Shift: Compared to the graph of √x, this graph is shifted one unit to the right and one unit up due to the “+1” added to the function and the “-1” in the radicand.

Soviet Joke: In Soviet Russia, square root shifts you! 🇷🇺

Identify the Radicand: In the function √(x-1) – 2, the radicand (the number under the square root) is x-1.
Set the Radicand ≥ 0: The square root of a negative number is not defined in the real number system, so x-1 must be greater than or equal to zero.
Solve for x: Solving the inequality x-1 ≥ 0 gives x ≥ 1.
Domain: The domain of √(x-1) – 2 is x ≥ 1.

Range: The range of √(x-1) – 2 is y ≥ -2. The graph starts at the point (1,-2) and extends infinitely upwards and to the right.
Curve Shape: The graph is a curve that starts at the point (1,-2) and extends infinitely to the right, getting gradually flatter as x increases.
End Behavior: As x approaches infinity, y also approaches -2, but at a decreasing rate.
Shift: Compared to the graph of √x, this graph is shifted one unit to the right and two units down due to the “+2” added to the function and the “-1” in the radicand.