The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$f\left(x\right)=(x+1{)}^2-1g\left(x\right)=(x+1{)}^2+1$
$h\left(x\right)=(x-1{)}^2+1j\left(x\right)=(x-1{)}^2-1$

Verified step by step guidance

1. Identify the vertex of the parabola from the graph. Here, the vertex is given as \(\left(\frac{1}{2}, 2\right)\).
2. Recall the vertex form of a quadratic function: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex. Substitute \(h = \frac{1}{2}\) and \(k = 2\) into the equation to get \(f(x) = a\left(x - \frac{1}{2}\right)^2 + 2\).
3. Use the y-intercept point \((0, 3)\) to find the value of \(a\). Substitute \(x = 0\) and \(f(x) = 3\) into the vertex form equation: \$3 = a\left(0 - \frac{1}{2}\right)^2 + 2$.
4. Simplify the equation to solve for \(a\): \$3 = a\left(\frac{1}{4}\right) + 2\(. Then isolate \)a\( by subtracting 2 from both sides and dividing by \)\frac{1}{4}$.
5. Once \(a\) is found, write the final quadratic function equation by substituting \(a\) back into the vertex form: \(f(x) = a\left(x - \frac{1}{2}\right)^2 + 2\).

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex and understand the graph's shape and position. In this problem, the vertex is given as (1/2, 2), which helps in writing the equation.

Using Points to Find the Quadratic Equation

To determine the quadratic function, you can substitute known points from the graph into the vertex form equation. For example, the y-intercept (0, 3) can be used to solve for the coefficient 'a' after substituting x = 0 and y = 3. This step is crucial to finalize the equation.

Graph Interpretation and Coordinate Identification

Understanding how to read the graph and identify key points like the vertex and intercepts is essential. The vertex indicates the parabola's minimum or maximum, and intercepts show where the graph crosses the axes. Accurate identification of these points is necessary to write the correct quadratic function.