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Describe the transformations between

y=√x and y= -√'x+1+5
Answer:
Part B
Describe the transformations between y=√x and y=-√(x+1+5) .
Answer:

Part C
Compare and contrast the transformations of the functions in parts A and B. What’s the same? What’s different?
Answer:

1 answer

Part A: Transformations from y = √x to y = -√(x + 1) + 5

  1. Horizontal Shift: The term inside the square root, \(x + 1\), indicates a horizontal shift. Since the \(+1\) moves the graph to the left by 1 unit, the function \(y = -√(x + 1)\) is the result of shifting \(y = √x\) left by 1 unit.

  2. Reflection: The negative sign in front of the square root, \(-√(x + 1)\), indicates a reflection across the x-axis. This transforms the graph from opening upward (as in y = √x) to opening downward.

  3. Vertical Shift: The \(+5\) outside the square root indicates a vertical shift. This moves the graph up by 5 units.

Summary of Transformations:

  • Shift left by 1 unit
  • Reflect across the x-axis
  • Shift up by 5 units

Part B: Transformations from y = √x to y = -√(x + 1 + 5)

  1. Horizontal Shift: The function \(x + 1 + 5\) simplifies to \(x + 6\). Therefore, the graph is shifted left by 6 units.

  2. Reflection: Similar to Part A, the negative sign in front of the square root indicates a reflection across the x-axis, transforming the upward opening to downward opening.

  3. Vertical Shift: Since the additional constants \(1 + 5 = 6\) are collectively inside the square root, when treated as \( -\sqrt{(x + 6)}\), it does not contribute to additional vertical displacement beyond reflection, but it’s effectively still acting as a vertical translation when combined with reflection.

Summary of Transformations:

  • Shift left by 6 units
  • Reflect across the x-axis

Part C: Comparison of Transformations

Similarities:

  1. Reflection Across x-axis: Both transformations involve reflecting the graph of \(y = √x\) across the x-axis, resulting in the downward-opening graphs.

  2. Horizontal Shift: Both transformations involve horizontal shifts, but they differ in magnitude and direction.

Differences:

  1. Horizontal Shift: In Part A, the graph is shifted left by 1 unit, while in Part B, it is shifted left by 6 units. This means the endpoint of the function will be at different x-values.

  2. Vertical Shift: Part A includes a vertical upward shift of 5 units after the reflection, while Part B does not have an explicit vertical shift; the reflection alone creates a different starting point compared to Part A.

Overall, the transformation from \(y=√x\) to \(y=-√(x + 1) + 5\) is more complex due to the combination of shifts and reflections, whereas the transformation to \(y=-√(x + 6)\) is simpler, involving mainly a leftward shift and reflection.