In Exercises 1–4, the graph of a quadratic function is given. Write the function's equation, selecting from the following options.

Verified step by step guidance

1. Identify the vertex of the quadratic function from the graph. The vertex is at (1/2, 2).
2. Recognize that the quadratic function can be written in vertex form: f(x) = a(x - h)^2 + k, where (h, k) is the vertex.
3. Substitute the vertex (1/2, 2) into the vertex form equation: f(x) = a(x - 1/2)^2 + 2.
4. Use another point on the graph to find the value of 'a'. For example, use the point (0, 3).
5. Substitute the point (0, 3) into the equation: 3 = a(0 - 1/2)^2 + 2 and solve for 'a'.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the general shape and properties of parabolas is essential for analyzing their equations.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards. For a quadratic function in vertex form, f(x) = a(x-h)² + k, the vertex is located at the point (h, k). In the given graph, the vertex is at (1/2, 2), which is crucial for writing the function's equation accurately.

Y-Intercept

The y-intercept of a function is the point where the graph intersects the y-axis, which occurs when x = 0. For quadratic functions, the y-intercept can be found by evaluating the function at x = 0, yielding the value of 'c' in the standard form f(x) = ax² + bx + c. In this case, the y-intercept is at (0, 3), providing a key point for determining the function's equation.