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 Which transformations are needed to graph the function h(x) = sqrt(-x) - 3 ? Choose the correct answer. A. The graph of y = sqrt(x) should be horizontally shifted to the right by 3 units and reflected about the x-axis. B. The graph of y = sqrt(x) should be vertically shifted up by 3 units and reflected about the x-axis. C. The graph of y = sqrt(x) should be vertically shifted down by 3 units and reflected about the y-axis. D. The graph of y = sqrt(x) should be horizontally shifted to the left by 3 units and reflected about the y-axis.

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 Which transformations are needed to graph the function h(x) = sqrt(-x) - 3 ? Choose the correct answer. A. The graph of y = sqrt(x) should be horizontally shifted to the right by 3 units and reflected about the x-axis. B. The graph of y = sqrt(x) should be vertically shifted up by 3 units and reflected about the x-axis. C. The graph of y = sqrt(x) should be vertically shifted down by 3 units and reflected about the y-axis. D. The graph of y = sqrt(x) should be horizontally shifted to the left by 3 units and reflected about the y-axis.
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. \[ h(x)=\sqrt{-x}-3 \] Which transformations are needed to graph the function $h(x)=\sqrt{-x}-3$ ? Choose the correct answer. A. The graph of $y=\sqrt{x}$ should be horizontally shifted to the right by 3 units and reflected about the $x$-axis. B. The graph of $y=\sqrt{x}$ should be vertically shifted up by 3 units and reflected about the $x$-axis. C. The graph of $y=\sqrt{x}$ should be vertically shifted down by 3 units and reflected about the $y$-axis. D. The graph of $y=\sqrt{x}$ should be horizontally shifted to the left by 3 units and reflected about the $y$-axis.
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Solution

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Solution Steps

Step 1: Identify the Base and Target Functions

The base function is $f(x) = sqrt(x)$ and the target function is $g(x) = sqrt(-x) - 3$.

Step 2: Determine the Transformations Needed
  • Vertical shift 3 units down
  • Reflection across the $y$-axis
Step 3: Apply the Transformations

The transformations are applied in the following order:

  1. Reflections
  2. Stretches/Compressions
  3. Translations (Shifts) After applying these transformations, we obtain the function $g(x)$ as described.

Final Answer:

After applying the specified transformations to $f(x)$, we get the function $g(x)$. This includes:

  • Vertical shift 3 units down
  • Reflection across the $y$-axis All transformations are applied with respect to the original function $f(x)$ to obtain $g(x)$, rounded to 0 decimal places where applicable.
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