Graph each function. See Examples 1 and 2. ƒ(x)=-(1/2)x^2

Verified step by step guidance

1. Step 1: Identify the type of function. The function ƒ(x)=-(1/2)x^2 is a quadratic function. Quadratic functions have the general form ƒ(x)=ax^2+bx+c, where a, b, and c are constants. In this case, a=-1/2, b=0, and c=0.
2. Step 2: Identify the vertex of the parabola. For a quadratic function in the form ƒ(x)=ax^2+bx+c, the vertex is at the point (-b/2a, ƒ(-b/2a)). In this case, since b=0 and a=-1/2, the vertex is at the point (0, ƒ(0)) = (0, 0).
3. Step 3: Identify the direction of the parabola. If a>0, the parabola opens upwards. If a<0, the parabola opens downwards. In this case, since a=-1/2, the parabola opens downwards.
4. Step 4: Plot the vertex and a few other points. For example, you could plot the points (-1, ƒ(-1)) = (-1, -1/2), (1, ƒ(1)) = (1, -1/2), and (2, ƒ(2)) = (2, -2).
5. Step 5: Draw the parabola. Connect the points with a smooth curve, making sure the parabola opens downwards and has its vertex at the origin.

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'. In this case, since 'a' is negative (-1/2), the parabola opens downwards.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on the direction it opens. For the function f(x) = -(1/2)x^2, the vertex is at the origin (0,0) because there are no linear or constant terms to shift it. The vertex is crucial for graphing as it helps determine the shape and position of the parabola.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of the function to create an accurate representation of its graph. For quadratic functions, key points include the vertex, intercepts, and additional points derived from substituting values into the function. Understanding these techniques is essential for visualizing the behavior of the function across its domain.