https://file.aiquickdraw.com/mathbot/file/1728446173293biallvif.jpg

The graph depicts a parabola that appears to be symmetric around the y-axis, suggesting that the equation of the curve follows the form of a quadratic equation:

\[
y = ax^2
\]

where \(a\) is the constant of proportionality (the coefficient of \(x^2\)).

### To estimate the constant of proportionality:
1. **Identify Points on the Parabola:**
   From the graph, some easily identifiable points are:
   - \((0, -4)\) (vertex of the parabola)
   - \((-2, 0)\)
   - \((2, 0)\)

2. **Using the Equation:**
   Since the parabola is symmetric around the y-axis and opens upwards, the general form is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. In this case, the vertex is at \((0, -4)\), so the equation becomes:

   \[
   y = a x^2 - 4
   \]

3. **Substitute a Point:**
   Let's substitute one of the points, \((2, 0)\), into the equation to find \(a\):

   \[
   0 = a(2)^2 - 4
   \]

   \[
   0 = 4a - 4
   \]

   Solving for \(a\):

   \[
   4a = 4 \quad \Rightarrow \quad a = 1
   \]

Thus, the constant of proportionality is **1**.

Would you like more details or have any further questions on this?

Here are some related questions:
1. How would the constant of proportionality change if the parabola opened downward?
2. What happens to the equation if the vertex of the parabola moves to a different point?
3. Can you generalize the steps to estimate \(a\) for any parabola given a graph?
4. How can we determine the axis of symmetry from the equation of a parabola?
5. What role does the constant of proportionality play in the shape of the parabola?

**Tip:** When dealing with quadratic functions, always identify key points like the vertex and x-intercepts to easily estimate the equation.

Given the graph, estimate the constant of proportionality.

Learn how to estimate the constant of proportionality from a parabola graph. This example includes identifying key points like the vertex and x-intercepts, using quadratic equations, and understanding symmetry. Suitable for students in Grades 9-10.

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