Graph each function. See Examples 1 and 2. g(x) = ½ x²

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Recognize that the function given is a quadratic function of the form \(g(x) = \frac{1}{2} x^{2}\), which is a parabola opening upwards because the coefficient of \(x^{2}\) is positive.

Identify the vertex of the parabola. Since there is no linear or constant term, the vertex is at the origin \((0,0)\).

Create a table of values by choosing several \(x\) values (both positive and negative), then calculate the corresponding \(g(x)\) values using the formula \(g(x) = \frac{1}{2} x^{2}\).

Plot the points from the table on the coordinate plane. Because the parabola is symmetric about the \(y\)-axis, points for positive and negative \(x\) values will be mirror images.

Draw a smooth curve through the plotted points to complete the graph of \(g(x) = \frac{1}{2} x^{2}\), ensuring the shape is a U-shaped parabola opening upwards.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Understanding Quadratic Functions

A quadratic function is a polynomial of degree two, typically written as f(x) = ax² + bx + c. Its graph is a parabola, which opens upward if a > 0 and downward if a < 0. In this case, g(x) = ½ x² is a parabola opening upward with vertex at the origin.

Graphing Parabolas

To graph a parabola, plot the vertex and several points by substituting x-values into the function. The shape is symmetric about the vertical axis through the vertex. For g(x) = ½ x², the parabola is wider than y = x² because the coefficient ½ compresses it vertically.

Effect of Coefficients on Graph Shape

The coefficient in front of x² affects the parabola's width and direction. A smaller positive coefficient (like ½) makes the parabola wider, while a larger coefficient makes it narrower. Understanding this helps in predicting and sketching the graph accurately.