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 by recalling 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. Identify the transformations applied to \(f(x)\) to get \(h(x) = -2(x+2)^{2} + 1\). Notice the expression inside the squared term is \((x + 2)\), which indicates a horizontal shift.
3. Apply the horizontal shift by moving the graph of \(f(x)\) left by 2 units, changing the vertex from \((0,0)\) to \((-2,0)\).
4. Next, observe the coefficient \(-2\) outside the squared term. The negative sign reflects the parabola across the x-axis (it opens downward), and the factor 2 vertically stretches the graph by a factor of 2.
5. Finally, apply the vertical shift by adding 1, moving the vertex up by 1 unit. The new vertex is at \((-2,1)\). Combine all transformations to sketch the graph of \(h(x)\).

Key Concepts

Standard Quadratic Function

The standard quadratic function is f(x) = x², which graphs as a parabola opening upwards with its vertex at the origin (0,0). Understanding this basic shape is essential because transformations are applied relative to this parent graph.

Transformations of Functions

Transformations include shifts, reflections, stretches, and compressions applied to the parent function. For h(x) = -2(x+2)² + 1, the graph shifts left by 2 units, reflects over the x-axis, vertically stretches by a factor of 2, and shifts up by 1 unit.

Vertex Form of a Quadratic Function

The vertex form is h(x) = a(x-h)² + k, where (h,k) is the vertex. This form makes it easier to identify transformations and graph the function by locating the vertex and understanding how 'a' affects the parabola's shape and direction.