Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note following Example 3.)

Verified step by step guidance

1. Identify the vertex of the quadratic function from the graph. Here, the vertex is given as (3, -9).
2. Recall the vertex form of a quadratic function: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
3. Substitute the vertex coordinates into the vertex form: \(f(x) = a(x - 3)^2 - 9\).
4. Use another point on the graph to find the value of \(a\). For example, if the graph passes through the point (0, 0), substitute \(x = 0\) and \(f(x) = 0\) into the equation and solve for \(a\).
5. Once \(a\) is found, write the final quadratic function in vertex form using the values of \(a\), \(h\), and \(k\).

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is 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 given as (3, -9), which helps in writing the function.

Determining the Value of 'a' in the Quadratic Function

The coefficient 'a' in the quadratic function affects the parabola's width and direction (upward if a > 0, downward if a < 0). To find 'a', use another point on the graph besides the vertex. This step is essential to fully define the quadratic equation.

Graph Interpretation and Coordinate Points

Understanding how to read the graph and extract key points like the vertex and other points is crucial. These points provide the necessary data to substitute into the quadratic equation to solve for unknowns and verify the function matches the graph.