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 \((-1, 7)\).
2. Recall the vertex form of a quadratic function: \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. Substitute \(h = -1\) and \(k = 7\) to get \(y = a(x + 1)^2 + 7\).
3. Use the y-intercept point to find the value of \(a\). The y-intercept is where \(x = 0\), and from the graph, the y-intercept is \(8\). Substitute \(x = 0\) and \(y = 8\) into the equation: \$8 = a(0 + 1)^2 + 7$.
4. Simplify the equation to solve for \(a\): \$8 = a(1)^2 + 7\( which simplifies to \)8 = a + 7\(. Then solve for \)a$ by subtracting 7 from both sides.
5. Write the final equation of the quadratic function by substituting the value of \(a\) back into the vertex form: \(y = a(x + 1)^2 + 7\).

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by f(x) = 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 (-1, 7), so the equation will be based on this point.

Using Points to Find the Leading Coefficient

To determine the value of 'a' in the vertex form, substitute another point on the graph into the equation. The y-intercept (0, 8) is a convenient point to use. By plugging x = 0 and y = 8 into the vertex form, you can solve for 'a', which controls the parabola's width and direction.

Graph Interpretation and Equation Writing

Interpreting the graph involves identifying key features like the vertex and intercepts, then translating these into an algebraic equation. Understanding how the graph's shape relates to the equation helps in selecting the correct function from given options. This skill is essential for connecting visual data to algebraic expressions.