Question

find an equation for the parabola.

Explanation:

Step1: Identify the vertex form

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From the graph, the vertex is at \((0, 1)\) (since it's on the y - axis, \(h = 0\) and \(k = 1\)). So the equation becomes \( y = a(x - 0)^2+1=a x^{2}+1\).

Step2: Find the value of \(a\)

We can use a point on the parabola. Let's take the point \((5, 3)\) (from the graph, when \(x = 5\), \(y = 3\)). Substitute \(x = 5\) and \(y = 3\) into the equation \(y=ax^{2}+1\):

$$
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$$

Step3: Write the final equation

Substitute \(a=\frac{2}{25}\), \(h = 0\) and \(k = 1\) into the vertex form. The equation of the parabola is \(y=\frac{2}{25}x^{2}+1\).

Answer:

\(y=\frac{2}{25}x^{2}+1\)