Question

\(g(x)=-2(x+1)^2-2\)

Short answer

The constants for the function are \(a=-2\), \(h=-1\), and \(k=-2\). The vertex of the parabola is at the point \((-1,-2)\) and the parabola opens downwards.

Step-by-step solution

STEP 1: Identify the Constants

In the given function \(g(x)=-2(x+1)^2-2\), the constants are as follows: \(a=-2\) (which will flip the parabola vertically and stretch it by a factor of 2), \(h=-1\) (which will shift the parabola 1 unit to the left), and \(k=-2\) (which will shift the parabola 2 units down).

STEP 2: Identify the Vertex of the Parabola

The vertex of a parabola in the form \(g(x) = a(x-h)^2 + k\) is at the point \((h,k)\). So for the function \(g(x)=-2(x+1)^2-2\), the vertex of the parabola is at the point \((-1,-2)\).

Determine Whether Parabola Opens Upwards or Downwards

Since the value of \(a\), which is -2, is less than zero, the parabola will open downwards.

Sketch the Parabola

Plot the vertex at \((-1,-2)\). Since the parabola opens downwards and is stretched by a factor of 2, points on the parabola will be twice as far from the vertex as they are in the basic \(y=x^2\) parabola. Hand draw or use a graphing tool to sketch the function.

Key concepts

Parabola Vertex Form

Understanding the vertex form of a quadratic function is crucial for easily identifying key features, such as the vertex. The vertex form of a quadratic function is given by:

• \(g(x) = a(x-h)^2 + k\)

Here:

• \(a\) determines the direction and the shape of the parabola,
• \((h, k)\) is the vertex of the parabola.

For the function \(g(x) = -2(x+1)^2-2\), you can directly see that the transformation is in vertex form. The vertex \((h, k)\) in this function is \((-1, -2)\), indicating the parabola's highest or lowest point. Knowing this form helps in quickly visualizing how the parabola is situated on the graph.

Transformations of Quadratic Functions

Transformations can move a quadratic function around the coordinate plane. They include shifting, reflecting, and stretching or compressing. Transformation of a quadratic function is based on the values of \(h\), \(k\), and \(a\) in the vertex form \(g(x) = a(x-h)^2 + k\).

In our function \(g(x)=-2(x+1)^2-2\):

• The term \(x+1\) indicates a horizontal shift to the left by 1 unit, because \(h = -1\).
• The \(-2\) outside the squared term shifts the graph down by 2 units as \(k = -2\).

After applying these shifts, the entire graph is relocated on the plane, effectively placing the vertex at the point \((-1, -2)\). These movements help in obtaining the correct position of the parabola on the graph.

Reflection and Stretching of Parabolas

When working with quadratics, the value of \(a\) in \(g(x)=a(x-h)^2+k\) plays a vital role in how the parabola looks. In the equation \(g(x) = -2(x+1)^2-2\), \(a = -2\) does two main things:

• It reflects the parabola. Since \(a\) is negative, the parabola opens downwards, making it an inverted "U".
• The magnitude of \(a\), which is 2 here, indicates vertical stretching. The parabola will be narrower on the graph compared to \(y = x^2\), as points move further away from the vertex at a faster rate, stretched by this factor of 2.

Both reflection and stretching are essential to predict how the graph appears without plotting every point manually. This understanding gives a visual guide of the parabola's behavior.