In Exercises 5–8, 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 parabola from the graph, which is the highest point (-3, 18).
2. Use the vertex form of a quadratic function: f(x) = a(x - h)^2 + k, where (h, k) is the vertex.
3. Substitute the vertex (-3, 18) into the vertex form: f(x) = a(x + 3)^2 + 18.
4. Use another point on the graph, such as (0, 9), to find the value of 'a'. Substitute x = 0 and f(x) = 9 into the equation: 9 = a(0 + 3)^2 + 18.
5. Solve the equation 9 = 9a + 18 for 'a' to find the value of '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 properties of parabolas, such as their vertex, axis of symmetry, and intercepts, is essential for analyzing their equations.

Vertex Form of a Quadratic

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 is particularly useful for identifying the vertex directly and understanding the transformations of the graph. By converting standard form to vertex form, one can easily determine the maximum or minimum value of the function, which is critical for solving related problems.

Using Points to Determine the Equation

To find the equation of a quadratic function from its graph, one can use known points on the curve. By substituting the coordinates of these points into the general form of the quadratic equation, a system of equations can be created to solve for the coefficients a, b, and c. This method allows for the precise formulation of the quadratic equation that corresponds to the given graph.