In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = -2(x+2)²+1

Verified step by step guidance

1. Start with the graph of the standard quadratic function \( f(x) = x^2 \), which is a parabola opening upwards with its vertex at the origin (0,0).
2. Apply a horizontal shift to the left by 2 units. This transformation changes the function to \( f(x) = (x+2)^2 \). The vertex of the parabola is now at (-2,0).
3. Apply a vertical stretch by a factor of 2 and reflect the graph across the x-axis. This transformation changes the function to \( f(x) = -2(x+2)^2 \). The parabola now opens downwards and is narrower than the original.
4. Apply a vertical shift upwards by 1 unit. This transformation changes the function to \( h(x) = -2(x+2)^2 + 1 \). The vertex of the parabola is now at (-2,1).
5. Combine all transformations to graph \( h(x) = -2(x+2)^2 + 1 \). The final graph is a downward-opening parabola with vertex at (-2,1), stretched vertically by a factor of 2.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The graph of a quadratic function is a parabola, which opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the basic shape and properties of the standard quadratic function, f(x) = x², is essential for applying transformations.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For quadratic functions, common transformations include vertical and horizontal shifts, vertical stretches or compressions, and reflections across the x-axis. In the given function h(x) = -2(x+2)² + 1, the transformations include a reflection over the x-axis, a vertical stretch by a factor of 2, and shifts to the left and up.

Vertex Form of Quadratic Functions

The vertex form of a quadratic function is expressed as f(x) = a(x-h)² + k, where (h, k) is the vertex of the parabola. This form makes it easier to identify the vertex and understand how transformations affect the graph. In the function h(x) = -2(x+2)² + 1, the vertex is located at (-2, 1), indicating the point where the parabola reaches its maximum value due to the negative coefficient.