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To determine the equation of the quadratic function \( g \) shown in the graph and find its intercepts with the coordinate axes, let's analyze the key points and features of the parabola.

### Step 1: Identify the Form of the Function
A quadratic function can generally be written as:
\[
g(x) = ax^2 + bx + c
\]
Alternatively, if we have the vertex and another point, we can use the vertex form:
\[
g(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex.

### Step 2: Find the Vertex
From the graph:
- The vertex appears to be at \( (-3, -4) \).
  - This suggests that the equation can take the form \( g(x) = a(x + 3)^2 - 4 \).

### Step 3: Identify Another Point on the Curve
Another clearly marked point is the y-intercept at \( (0, 2) \). Substituting this point into the equation will help us find \( a \):
\[
2 = a(0 + 3)^2 - 4
\]
\[
2 = 9a - 4
\]
\[
9a = 6
\]
\[
a = \frac{2}{3}
\]

### Step 4: Write the Equation
Now that we have \( a = \frac{2}{3} \), the equation of the function \( g(x) \) is:
\[
g(x) = \frac{2}{3}(x + 3)^2 - 4
\]

### Step 5: Find the Intercepts
1. **Y-Intercept**: We've already identified this as \( (0, 2) \).
2. **X-Intercepts**: Set \( g(x) = 0 \) and solve for \( x \):
   \[
   0 = \frac{2}{3}(x + 3)^2 - 4
   \]
   \[
   \frac{2}{3}(x + 3)^2 = 4
   \]
   \[
   (x + 3)^2 = 6
   \]
   \[
   x + 3 = \pm \sqrt{6}
   \]
   \[
   x = -3 \pm \sqrt{6}
   \]
   So the x-intercepts are \( x = -3 + \sqrt{6} \) and \( x = -3 - \sqrt{6} \).

### Summary
- The equation of the quadratic function is:
  \[
  g(x) = \frac{2}{3}(x + 3)^2 - 4
  \]
- The intercepts are:
  - Y-intercept at \( (0, 2) \)
  - X-intercepts at \( \left( -3 + \sqrt{6}, 0 \right) \) and \( \left( -3 - \sqrt{6}, 0 \right) \)

Would you like more details or have any questions?

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#### Related Questions:
1. How can you convert a quadratic function from vertex form to standard form?
2. What are some other methods for finding the intercepts of a quadratic function?
3. How does changing the value of \( a \) affect the shape of a parabola?
4. What is the axis of symmetry for this parabola, and how can it be derived?
5. How can the quadratic formula be used to find the roots of a function in standard form?

#### Tip:
When you know the vertex and another point, using the vertex form of a quadratic equation is a quick and efficient way to find the function's rule.

Determine la regla de correspondencia de la función cuadrática g mostrada en la figura y los puntos que intercepta su gráfico con los ejes coordenados.

This problem involves finding the equation of a quadratic function based on its graph and calculating the intercepts with the x- and y-axes. The solution explores algebraic techniques using vertex form and intercepts, making it suitable for students in Grades 9-12.

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