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The functions f and g are graphed in the same rectangular coordinate​ system, shown to the right. If g is obtained from f through a sequence of​ transformations, find an equation for g.

The image shows two functions, \( f(x) = x^2 \) and another function \( g(x) \), which is a transformed version of \( f(x) \). The function \( f(x) = x^2 \) is a standard parabola opening upwards, while \( g(x) \) appears to be a parabola that has been reflected, shifted, and possibly stretched or compressed.

To determine the equation of \( g(x) \), we need to analyze the transformations that have been applied to \( f(x) = x^2 \). 

### Step 1: Reflection
The graph of \( g(x) \) is opening downwards, which indicates a reflection over the x-axis. This means that the function has been multiplied by \(-1\). So, the equation becomes:
\[
g(x) = -f(x)
\]

### Step 2: Horizontal Shift
The vertex of \( g(x) \) appears to be shifted from the origin to the right. If the vertex of \( g(x) \) is at \( (h, k) \), then the equation will include a horizontal shift:
\[
g(x) = -f(x-h)
\]
Based on the graph, the vertex of \( g(x) \) seems to be at \( (2, 0) \). So, the equation becomes:
\[
g(x) = -(x-2)^2
\]

### Step 3: Vertical Shift
Next, there is a vertical shift. The function \( g(x) \) appears to have been shifted downward by 4 units. So, the final equation is:
\[
g(x) = -(x-2)^2 - 4
\]

Thus, the equation for \( g(x) \) is:
\[
\boxed{g(x) = -(x-2)^2 - 4}
\]

Would you like more details or have any questions? Here are some related questions you might consider:

1. What would be the effect of shifting \( f(x) \) to the left instead of the right?
2. How does the equation of a function change if it is reflected over the y-axis instead of the x-axis?
3. What is the general form of the equation for a parabola that has been reflected, shifted, and stretched?
4. How do vertical stretches and compressions affect the graph of \( f(x) = x^2 \)?
5. If \( g(x) = -(x-2)^2 + 3 \), what transformations have been applied to \( f(x) = x^2 \)?

**Tip:** When analyzing transformations, always start with reflections and shifts before considering stretches and compressions.

Explore how to find the equation for g(x) from f(x) = x^2 through transformations. Learn about reflections, horizontal and vertical shifts, and their effects on quadratic functions. Suitable for Grades 10-12.

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