Questions: Explain how the graph of g is obtained from the graph of f. (a) f(x) = sqrt(x), g(x) = -sqrt(x) + 7

Explain how the graph of g is obtained from the graph of f. (a) f(x) = sqrt(x), g(x) = -sqrt(x) + 7
Transcript text: Explain how the graph of $g$ is obtained from the graph of $f$. (a) $f(x)=\sqrt{x}, \quad g(x)=-\sqrt{x}+7$
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Solution

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Solution Steps

To obtain the graph of \( g(x) \) from the graph of \( f(x) \):

  1. Start with the graph of \( f(x) = \sqrt{x} \).
  2. Reflect the graph of \( f(x) \) over the x-axis to get \( -\sqrt{x} \).
  3. Shift the resulting graph upward by 7 units to get \( g(x) = -\sqrt{x} + 7 \).
Step 1: Identify the Base Function

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

Step 2: Analyze the Transformation

The function \( g(x) = -\sqrt{x} + 7 \) can be broken down into two transformations applied to \( f(x) \).

  1. Vertical Reflection: The term \(-\sqrt{x}\) indicates a reflection of \( f(x) \) across the x-axis.
  2. Vertical Translation: The term \( +7 \) indicates a vertical shift upwards by 7 units.
Step 3: Apply the Transformations
  1. Vertical Reflection: Reflecting \( f(x) = \sqrt{x} \) across the x-axis gives \( -\sqrt{x} \).
  2. Vertical Translation: Shifting \( -\sqrt{x} \) upwards by 7 units results in \( -\sqrt{x} + 7 \).

Final Answer

The graph of \( g(x) = -\sqrt{x} + 7 \) is obtained from the graph of \( f(x) = \sqrt{x} \) by reflecting it across the x-axis and then shifting it upwards by 7 units.

\[ \boxed{\text{Reflect across the x-axis and shift upwards by 7 units}} \]

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