https://file.aiquickdraw.com/mathbot/file/172892583683421ris01z.jpg

The problem asks to determine the equation of the quadratic function \( g \), as well as the intercepts with the axes, based on the provided graph.

### Step 1: Identify key points from the graph
From the image:
- The graph of the function is a parabola that opens upwards.
- The vertex of the parabola appears to be at approximately \( (1, -2) \), suggesting this is a minimum point.
- The parabola intersects the y-axis at a point approximately \( y = -2 \).
- Another important point given on the graph is around \( (4, 5) \).

### Step 2: General form of a quadratic function
The general form of a quadratic function is:
\[
g(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola. From the graph, the vertex appears to be at \( (1, -2) \), so we substitute \( h = 1 \) and \( k = -2 \) to get:
\[
g(x) = a(x - 1)^2 - 2
\]

### Step 3: Use another point to find \( a \)
We can now substitute another known point to solve for the value of \( a \). From the graph, the point \( (4, 5) \) lies on the curve. Substituting \( x = 4 \) and \( y = 5 \) into the equation:
\[
5 = a(4 - 1)^2 - 2
\]
\[
5 = a(3)^2 - 2
\]
\[
5 = 9a - 2
\]
\[
7 = 9a
\]
\[
a = \frac{7}{9}
\]

### Step 4: Write the equation of the function
Thus, the equation of the quadratic function is:
\[
g(x) = \frac{7}{9}(x - 1)^2 - 2
\]

### Step 5: Find the intercepts
- **Y-intercept**: Set \( x = 0 \) in the equation:
\[
g(0) = \frac{7}{9}(0 - 1)^2 - 2 = \frac{7}{9} - 2 = -1.22
\]
Thus, the y-intercept is approximately \( (0, -1.22) \).

- **X-intercepts**: Set \( g(x) = 0 \) and solve for \( x \):
\[
0 = \frac{7}{9}(x - 1)^2 - 2
\]
\[
\frac{7}{9}(x - 1)^2 = 2
\]
\[
(x - 1)^2 = \frac{18}{7}
\]
\[
x - 1 = \pm \sqrt{\frac{18}{7}} \approx \pm 1.6
\]
\[
x = 1 \pm 1.6
\]
So, the x-intercepts are approximately \( x = 2.6 \) and \( x = -0.6 \).

### Final Results:
- The quadratic equation is \( g(x) = \frac{7}{9}(x - 1)^2 - 2 \).
- Y-intercept: \( (0, -1.22) \).
- X-intercepts: approximately \( (-0.6, 0) \) and \( (2.6, 0) \).

Do you have any questions or would you like further details?

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### Related Questions:
1. What is the significance of the vertex in the graph of a quadratic function?
2. How can we use symmetry to predict the intercepts of a quadratic function?
3. Can you calculate the roots of the quadratic equation using the quadratic formula?
4. How does the value of \( a \) in the equation affect the shape of the parabola?
5. What is the relationship between the axis of symmetry and the vertex of a quadratic function?

### Tip:
When finding intercepts of quadratic functions, setting \( y = 0 \) gives the x-intercepts, while setting \( x = 0 \) gives the y-intercept.

Determine the equation of the quadratic function g shown in the figure and the points where its graph intercepts the coordinate axes.

Solve for the quadratic function g from a given graph, and determine its x- and y-intercepts. The problem covers the vertex form of a parabola and finding intercepts for students in grades 9-11.

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