Graph each function. See Examples 1 and 2. ƒ(x)=-(1/2)x²

Verified step by step guidance

1. Identify the type of function given. The function ƒ(x) = -\(\frac{1}{2}\)x^2 is a quadratic function, which graphs as a parabola.
2. Note the coefficient of x^2, which is -\(\frac{1}{2}\). Since it is negative, the parabola opens downward, and the factor \(\frac{1}{2}\) affects the width, making it wider than the standard parabola y = -x^2.
3. Find the vertex of the parabola. For a function in the form ƒ(x) = ax^2 + bx + c, the vertex is at x = -\(\frac{b}{2a}\). Here, b = 0, so the vertex is at x = 0. Substitute x = 0 into the function to find the y-coordinate of the vertex.
4. Create a table of values by choosing x-values around the vertex (for example, x = -2, -1, 0, 1, 2), then calculate the corresponding y-values using the function ƒ(x) = -\(\frac{1}{2}\)x^2.
5. Plot the points from the table on the coordinate plane, mark the vertex, and draw a smooth curve through these points to complete the graph of the parabola.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two, typically written as f(x) = ax^2 + bx + c. Its graph is a parabola, which opens upward if a > 0 and downward if a < 0. Understanding the shape and properties of quadratic functions is essential for graphing them accurately.

Effect of the Leading Coefficient

The leading coefficient (the number multiplying x^2) affects the parabola's direction and width. A negative coefficient, like -1/2, causes the parabola to open downward, while its absolute value determines how wide or narrow the parabola is compared to the standard y = x^2.

Graphing Techniques for Quadratics

Graphing a quadratic involves plotting the vertex, axis of symmetry, and several points on either side. For f(x) = -(1/2)x^2, the vertex is at the origin (0,0), and symmetry about the y-axis helps in plotting points efficiently to sketch the parabola.