Questions: Graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function. Be sure to show at least three key points. Find the domain and the range of the function. h(x) = sqrt(-x) - 3

Graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function. Be sure to show at least three key points. Find the domain and the range of the function. h(x) = sqrt(-x) - 3
Transcript text: Graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function. Be sure to show at least three key points. Find the domain and the range of the function. \[ h(x)=\sqrt{-x}-3 \]
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Solution

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Solution Steps

Step 1: Identify the basic function

The basic function is \( f(x) = \sqrt{x} \).

Step 2: Apply transformations

The given function is \( h(x) = \sqrt{-x} - 3 \). This involves:

  • Reflecting \( \sqrt{x} \) over the y-axis to get \( \sqrt{-x} \).
  • Shifting the graph down by 3 units.
Step 3: Determine key points

For the basic function \( f(x) = \sqrt{x} \), key points are:

  • \( (0, 0) \)
  • \( (1, 1) \)
  • \( (4, 2) \)

Applying the transformations:

  • Reflecting over the y-axis: \( (0, 0) \), \( (-1, 1) \), \( (-4, 2) \)
  • Shifting down by 3 units: \( (0, -3) \), \( (-1, -2) \), \( (-4, -1) \)
Step 4: Find the domain and range
  • Domain: \( x \leq 0 \)
  • Range: \( y \geq -3 \)

Final Answer

The function \( h(x) = \sqrt{-x} - 3 \) is obtained by reflecting \( \sqrt{x} \) over the y-axis and then shifting it down by 3 units. The key points are \( (0, -3) \), \( (-1, -2) \), and \( (-4, -1) \). The domain is \( x \leq 0 \) and the range is \( y \geq -3 \).

{"axisType": 3, "coordSystem": {"xmin": -5, "xmax": 1, "ymin": -5, "ymax": 1}, "commands": ["y = sqrt(-x) - 3"], "latex_expressions": ["$h(x) = \\sqrt{-x} - 3$"]}

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