For the given transformations, we will analyze the transformations of two functions and express them using terms such as dilation, translation, and reflection.

Part (a)

The base function is $y = x^2$. We need to describe the sequence of transformations that converts $y = x^2$ into each of the following forms:

i. $y = 2(x + 1)^2 + 3$

• Horizontal translation: The term $(x + 1)^2$ indicates a translation 1 unit to the left.
• Vertical dilation: The factor of $2$ indicates a vertical stretch by a factor of 2.
• Vertical translation: The $+3$ outside the squared term shifts the graph up by 3 units.

Thus, the transformation sequence is:

1. Translate 1 unit to the left.
2. Vertically stretch by a factor of 2.
3. Translate 3 units upward.

ii. $y = -4(x + 1)^2 + 2$

• Horizontal translation: $(x + 1)^2$ indicates a shift 1 unit to the left.
• Vertical dilation: The factor $-4$ indicates a vertical stretch by a factor of 4 and a reflection across the x-axis (due to the negative sign).
• Vertical translation: The $+2$ shifts the graph up by 2 units.

Thus, the transformation sequence is:

1. Translate 1 unit to the left.
2. Reflect across the x-axis and vertically stretch by a factor of 4.
3. Translate 2 units upward.

iii. $y = 2(x + 1)^2 - 2$

• Horizontal translation: $(x + 1)^2$ shifts the graph 1 unit to the left.
• Vertical dilation: The factor $2$ indicates a vertical stretch by a factor of 2.
• Vertical translation: The $-2$ shifts the graph down by 2 units.

Thus, the transformation sequence is:

1. Translate 1 unit to the left.
2. Vertically stretch by a factor of 2.
3. Translate 2 units downward.

Part (b)

The base function is $y = \frac{1}{x}$. Now we need to transform it into each of the following forms:

i. $y = \frac{2}{x + 3}$

• Horizontal translation: The $x + 3$ in the denominator translates the graph 3 units to the left.
• Vertical dilation: The factor of $2$ in the numerator indicates a vertical stretch by a factor of 2.

Thus, the transformation sequence is:

1. Translate 3 units to the left.
2. Vertically stretch by a factor of 2.

ii. $y = \frac{1}{x + 3} + 2$

• Horizontal translation: The $x + 3$ in the denominator translates the graph 3 units to the left.
• Vertical translation: The $+2$ shifts the graph up by 2 units.

Thus, the transformation sequence is:

1. Translate 3 units to the left.
2. Translate 2 units upward.

iii. $y = \frac{1}{x - 3} - 2$

• Horizontal translation: The $x - 3$ in the denominator translates the graph 3 units to the right.
• Vertical translation: The $-2$ shifts the graph down by 2 units.

Thus, the transformation sequence is:

1. Translate 3 units to the right.
2. Translate 2 units downward.

Would you like further details or clarifications on any of these transformations?

Here are 5 additional questions related to this topic:

1. What is the general rule for translating a quadratic function horizontally and vertically?
2. How does a negative coefficient affect the graph of a function?
3. What effect does a vertical dilation have on the steepness of a function's graph?
4. How does the horizontal shift in a reciprocal function differ from a quadratic function?
5. What is the transformation sequence for reflecting a function across both the x-axis and the y-axis?

Tip: When identifying transformations, always analyze the expression inside parentheses for horizontal shifts and outside for vertical transformations.