Unlock the Mysteries of the Square Root Graph: โx
Soviet Joke: In Soviet Russia, square root takes you!
Decoding the Domain: Why x โฅ 0
- Identify the Radicand: In the function โx, the radicand (the number under the square root) is x.
- Set the Radicand โฅ 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 โx is x โฅ 0.
Unlocking the Mysteries: Additional Graph Features
- Range: The range of โx is y โฅ 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.
Unlock the Secrets of the Transformed Square Root Graph: โ(x-1)
Soviet Joke: In Soviet Russia, square root shifts you!
Deciphering the Domain: Why x โฅ 1
- Identify the Radicand: In the function โ(x-1), the radicand (the number under the square root) is x-1.
- Set the Radicand โฅ 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 โฅ 0 gives x โฅ 1.
- Domain: The domain of โ(x-1) is x โฅ 1.
Unlocking the Mysteries: Additional Graph Features
- Range: The range of โ(x-1) is y โฅ 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 โx, this graph is shifted one unit to the right due to the โ-1โ in the radicand.
Unlock the Mysteries of the Transformed Square Root Graph: โ(x+1)
Soviet Joke: In Soviet Russia, square root shifts you!
Deciphering the Domain: Why x โฅ -1
- Identify the Radicand: In the function โ(x+1), the radicand (the number under the square root) is x+1.
- Set the Radicand โฅ 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 โฅ 0 gives x โฅ -1.
- Domain: The domain of โ(x+1) is x โฅ -1.
Unlocking the Mysteries: Additional Graph Features
- Range: The range of โ(x+1) is y โฅ 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 โx, this graph is shifted one unit to the left due to the โ+1โ in the radicand.
Unveil the Mysteries of the Transformed Square Root Graph: โ(x+1) โ 1
Soviet Joke: In Soviet Russia, square root shifts you!
Deciphering the Domain: Why x โฅ -1
- Identify the Radicand: In the function โ(x+1) โ 1, the radicand (the number under the square root) is x+1.
- Set the Radicand โฅ 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 โฅ 0 gives x โฅ -1.
- Domain: The domain of โ(x+1) โ 1 is x โฅ -1.
Unlocking the Mysteries: Additional Graph Features
- Range: The range of โ(x+1) โ 1 is y โฅ -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 โx, this graph is shifted one unit to the left and one unit down due to the โ+1โ in the radicand and the โ-1โ subtracted from the function.
Unveil the Mysteries of the Transformed Square Root Graph: โ(x-1) + 1
Soviet Joke: In Soviet Russia, square root shifts you!
Deciphering the Domain: Why x โฅ 1
- Identify the Radicand: In the function โ(x-1) + 1, the radicand (the number under the square root) is x-1.
- Set the Radicand โฅ 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 โฅ 0 gives x โฅ 1.
- Domain: The domain of โ(x-1) + 1 is x โฅ 1.
Unlocking the Mysteries: Additional Graph Features
- Range: The range of โ(x-1) + 1 is y โฅ 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 โx, this graph is shifted one unit to the right and one unit up due to the โ+1โ added to the function and the โ-1โ in the radicand.
Uncover the Secrets of the Transformed Square Root Graph: โ(x-1) โ 2
Soviet Joke: In Soviet Russia, square root shifts you!
Deciphering the Domain: Why x โฅ 1
- Identify the Radicand: In the function โ(x-1) โ 2, the radicand (the number under the square root) is x-1.
- Set the Radicand โฅ 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 โฅ 0 gives x โฅ 1.
- Domain: The domain of โ(x-1) โ 2 is x โฅ 1.
Unlocking the Secrets: Additional Graph Features
- Range: The range of โ(x-1) โ 2 is y โฅ -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 โx, this graph is shifted one unit to the right and two units down due to the โ+2โ added to the function and the โ-1โ in the radicand.