In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)^2+5

Verified step by step guidance

1. Identify the standard form of a quadratic function, which is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
2. Compare the given function \( f(x) = -2(x+1)^2 + 5 \) with the standard form to identify the values of \(a\), \(h\), and \(k\).
3. Notice that \(a = -2\), \(h = -1\), and \(k = 5\) from the function \( f(x) = -2(x+1)^2 + 5 \).
4. Recognize that the vertex \((h, k)\) of the parabola is \((-1, 5)\).
5. Conclude that the coordinates of the vertex for the given quadratic function are \((-1, 5)\).

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the standard form and vertex form of quadratic functions is essential for analyzing their properties.

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). Identifying the vertex is crucial for graphing the parabola and understanding its maximum or minimum value.

Transformation of Functions

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the given function f(x) = -2(x+1)^2 + 5, the expression (x+1) indicates a horizontal shift to the left by 1 unit, while the -2 indicates a vertical stretch and reflection over the x-axis. Recognizing these transformations helps in accurately determining the vertex and overall shape of the parabola.