https://file.aiquickdraw.com/mathbot/file/1725275186166qzczo0ew.jpg

Какая формула соответствует этому графику?

The graph you uploaded appears to represent a quadratic function, which generally has the form:

\[
y = ax^2 + bx + c
\]

From the image, the graph is a parabola that opens upwards, which suggests that the coefficient \(a\) is positive. To determine the exact formula, we need to identify key points on the graph, such as the vertex and the roots (if they are visible), or any other points through which the curve passes.

### Observations:
1. **Vertex**: The vertex seems to be around \( (3, 0) \), which is a point on the x-axis.
2. **Roots**: The parabola intersects the x-axis at \(x = 3\).

### Vertex Form:
Given that the vertex of the parabola is at \( (3, 0) \), the equation of the parabola can be written in the vertex form:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola.

Substituting the vertex \( (3, 0) \):
\[
y = a(x - 3)^2
\]

Since the parabola opens upwards and appears symmetrical around \( x = 3 \), this equation is likely the correct one. To find the value of \( a \), we would need another point on the parabola. However, without additional points, we can assume \( a = 1 \) for simplicity:

\[
y = (x - 3)^2
\]

This is a simplified form of the quadratic equation that matches the observed graph.

Would you like more details or have any questions?

### Further Questions:
1. How would the graph change if \( a \) were negative?
2. What happens to the graph if we shift the vertex to a different point?
3. Can you determine the roots if \( a \) was any other positive number?
4. How would adding a linear term \( bx \) affect the graph?
5. What is the significance of the discriminant in determining the shape of the parabola?

### Tip:
Always check the vertex and the direction of the parabola's opening to quickly identify the basic form of the quadratic equation.

Explore the quadratic equation y = ax^2 + bx + c and vertex form y = a(x - h)^2 + k through detailed analysis of a specific graph. Learn how to identify key points like the vertex and roots to determine the equation of the parabola.

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