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

Verified step by step guidance

1. Identify the type of function given. Since \(g(x) = \frac{1}{2}x^2\) is a quadratic function, its graph will be a parabola.
2. Determine the key features of the parabola. The coefficient \(\frac{1}{2}\) in front of \(x^2\) affects the width and direction of the parabola. Since it is positive, the parabola opens upward, and because \(\frac{1}{2}\) is less than 1, the parabola will be wider than the standard \(x^2\) parabola.
3. Find the vertex of the parabola. For \(g(x) = a x^2 + bx + c\), 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 vertex point: \(g(0) = \frac{1}{2} \times 0^2\).
4. Create a table of values by choosing several \(x\) values (both positive and negative), then calculate the corresponding \(g(x)\) values using the function \(g(x) = \frac{1}{2}x^2\). This will give you points to plot on the graph.
5. Plot the vertex and the points from your table on the coordinate plane, then 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 parabolas is essential for graphing quadratic functions.

Effect of the Leading Coefficient

The leading coefficient 'a' in a quadratic function affects the width and direction of the parabola. If |a| < 1, the parabola is wider than the standard y = x^2; if |a| > 1, it is narrower. For g(x) = (1/2)x^2, the parabola opens upward and is wider than y = x^2.

Plotting Points and Symmetry

To graph a quadratic function, plot key points by substituting x-values and calculating corresponding y-values. Quadratic graphs are symmetric about the vertical axis through the vertex, making it easier to plot points on one side and reflect them on the other.