https://file.aiquickdraw.com/mathbot/file/1727054609035bccuc0qk.jpg

Let's analyze the graph and the questions provided in the image to solve each part.

### Part (a)
**State the transformations, in correct order, involved to generate the graph of \( y = g(x) \) from the graph of \( y = f(x) \):**

From the graph, we can identify the following transformations between \( f(x) \) (thin red line) and \( g(x) \) (thick purple line):
1. **Horizontal Stretch**: The graph \( f(x) \) seems to be horizontally stretched by a factor of 4.
2. **Vertical Stretch**: There is a vertical stretch by a factor of 2.
3. **Horizontal Shift (Translation)**: The graph is shifted 6 units to the right.
4. **Vertical Shift (Translation)**: The graph is shifted 3 units up.

Thus, the transformations are:
- Horizontal stretch by a factor of 4.
- Vertical stretch by a factor of 2.
- Horizontal shift by 6 units to the right.
- Vertical shift by 3 units up.

### Part (b)
**Express the transformations using mapping notation:**

Mapping notation describes how each coordinate \( (x, y) \) changes based on the transformations. Using the transformations described above:

\[
(x, y) \rightarrow \left( 4x + 6, 2y + 3 \right)
\]

This means:
- Multiply the \( x \)-coordinate by 4 and shift it 6 units to the right.
- Multiply the \( y \)-coordinate by 2 and shift it 3 units up.

### Part (c)
**Write the equation of the transformed function \( y = g(x) \):**

Starting with \( f(x) \), apply the transformations step by step:
1. A horizontal stretch by 4: Replace \( x \) by \( \frac{x}{4} \).
2. A vertical stretch by 2: Multiply the function by 2.
3. A horizontal shift by 6: Replace \( x \) by \( x - 6 \).
4. A vertical shift by 3: Add 3 to the function.

Therefore, if the original function is \( y = f(x) \), the transformed function is:

\[
y = g(x) = 2f\left(\frac{x-6}{4}\right) + 3
\]

### Part (d)
**Select the smallest \( x \)-intercept on the graph of \( y = f(x) \). Apply mapping notation to determine the exact value of the corresponding point on \( y = g(x) \):**

From the graph, it appears that the smallest \( x \)-intercept of \( f(x) \) is approximately \( x = -2 \).

Using the mapping notation from part (b):
\[
(x, y) \rightarrow \left( 4x + 6, 2y + 3 \right)
\]

Substitute \( x = -2 \) into the mapping rule for \( x \):

\[
4(-2) + 6 = -8 + 6 = -2
\]

Thus, the corresponding \( x \)-intercept for \( g(x) \) is \( x = -2 \).

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Would you like further details on any part of this solution?

### 5 Relative Questions:
1. How does a horizontal stretch affect the graph of a function geometrically?
2. What would happen if the vertical stretch was instead a vertical compression by \( \frac{1}{2} \)?
3. How can you verify the vertical and horizontal shifts using specific points on the graph?
4. What transformations are required to reflect \( f(x) \) across the \( y \)-axis?
5. How does applying multiple transformations to a function affect the domain and range?

### Tip:
When writing the equation of a transformed function, always apply the transformations in the correct order: stretches/compressions, then shifts.

Consider the graph where y = f(x) is represented by the thin red line and y = g(x) is represented by the thick purple line. The problem asks to analyze the transformations between these graphs.

Analyze the transformations between f(x) and g(x) involving horizontal and vertical stretches, shifts, and the use of mapping notation. Learn to apply these transformations to find the equation of the transformed function and the smallest x-intercept.

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