To answer Part A, let's break down the key concepts.

Transformations of Parent Functions:

For an exponential function, the general form is:

$f(x) = a \cdot b^{(x - h)} + k$

Where:

• a affects vertical stretch or compression and reflects the graph across the x-axis if negative.
• b is the base of the exponential function.
• h translates the graph horizontally (right if positive, left if negative).
• k translates the graph vertically (up if positive, down if negative).

Analyzing the Responses:

1. Response 1: "a vertical stretch by a factor of 4, a translation right 2 units, and a translation up 3 units"

   • This suggests:
     • Vertical Stretch by 4: This would imply a vertical stretch factor of 4, which means the coefficient a is 4.
     • Translation Right by 2: This means h = 2, so the graph shifts 2 units to the right.
     • Translation Up by 3: This means k = 3, so the graph shifts 3 units up.

2. Response 2: "a vertical stretch by a factor of __, a translation right __ units, and a translation down __ units"

   • Incomplete information. The vertical stretch factor, translation right, and translation down values need to be filled in.

3. Response 3: "a vertical stretch by a factor of 1 fifth, a translation left 2 units, and a translation up 3 units"

   • This suggests:
     • Vertical Stretch by 1/5: This would imply a vertical compression by a factor of 5 (since a value less than 1 is a compression).
     • Translation Left by 2: This means h = -2, so the graph shifts 2 units to the left.
     • Translation Up by 3: This means k = 3, so the graph shifts 3 units up.

4. Response 4: "a vertical stretch by a factor of __, a translation left __ units, and a translation up __ units"

   • Again, this is an incomplete response where the exact values for vertical stretch, translation left, and translation up should be specified.

5. Response 5: "a vertical stretch by a factor of 1 fifth, a translation left 2 units, and a translation up 3 units"

   • This suggests:
     • Vertical Stretch by 1/5: This implies a compression by a factor of 5.
     • Translation Left by 2: h = -2, so the graph shifts 2 units to the left.
     • Translation Up by 3: k = 3, so the graph shifts 3 units up.

Conclusion:

From the complete responses, Response 1 and Response 3 seem to provide fully explained transformations. Depending on the context of the problem, if the function has a vertical stretch of 4, translation to the right by 2, and up by 3, then Response 1 would be the correct choice.

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