Let's analyze and compare the two functions provided:
- Function Definitions:
- Function 1: \( y = \sqrt{x} \)
- Function 2: \( y = -\sqrt{x} + 1 + 5 \)
- Function 3: \( y = -(\sqrt{x} + 1 + 5) \)
Characteristics of Each Function
Function 1: \( y = \sqrt{x} \)
- Domain: \( x \geq 0 \) (since we only consider the square root of non-negative numbers)
- Range: \( y \geq 0 \) (output is always non-negative)
- Shape: This function represents a curve that starts at the origin (0, 0) and increases slowly and continuously towards the right, forming the right half of a parabola.
Function 2: \( y = -\sqrt{x} + 1 + 5 \)
- This can be simplified to \( y = -\sqrt{x} + 6 \).
- Domain: \( x \geq 0 \)
- Range: \( y \leq 6 \) (the highest point occurs when \( x = 0 \), giving \( y = 6 \), and decreases as \( x \) increases)
- Shape: This function is the reflection of \( y = \sqrt{x} \) across the x-axis, shifted up by 6 units. It starts at \( (0, 6) \) and decreases as \( x \) increases, creating a downward-opening curve.
Function 3: \( y = -(\sqrt{x} + 1 + 5) \)
- This can be simplified to \( y = -\sqrt{x} - 6 \).
- Domain: \( x \geq 0 \)
- Range: \( y \leq -6 \) (the lowest point occurs when \( x = 0 \), giving \( y = -6 \), and decreases as \( x \) increases)
- Shape: This function is also the reflection of \( y = \sqrt{x} \) across the x-axis, but shifted downward by 6 units. It starts at \( (0, -6) \) and decreases further as \( x \) increases.
Comparison Summary
- Domains: All three functions have the same domain of \( x \geq 0 \).
- Range:
- \( y = \sqrt{x} \): \( y \geq 0 \)
- \( y = -\sqrt{x} + 6 \): \( y \leq 6 \)
- \( y = -(\sqrt{x} + 1 + 5) \): \( y \leq -6 \)
- Shapes:
- \( y = \sqrt{x} \) is an upward-opening curve starting at the origin.
- \( y = -\sqrt{x} + 6 \) is a downward-opening curve starting at \( (0, 6) \).
- \( y = -(\sqrt{x} + 1 + 5) \) is also a downward-opening curve starting at \( (0, -6) \).
Conclusively
- Functions 2 and 3 are downward-opening transformations of the square root function, reflecting its basic shape.
- Function 1 is the original square root function, increasing positively.
- Functions 2 and 3 represent various shifts and reflections of the original function in different directions.
1 answer