Question

\(g(x)=\frac{1}{3} x^2\)

Short answer

The value of \(g(2)\) is \(\frac{4}{3}\).

Step-by-step solution

Understand the function

The function \(g(x) = \frac{1}{3}x^2\) is in the form of \(y = ax^2\), which is a quadratic function. Here, \(a = \frac{1}{3}\) is the coefficient, and \(x\) is the variable. This function produces a parabola graphic in the cartesian plane.

Substitute the value of \(x\) to find \(g(x)\)

If we want to find the value of \(g(x)\) for any particular value of \(x\), we substitute the given \(x\) value into the function. For example, to find \(g(2)\), we substitute \(2\) for \(x\) in the expression and calculate the result.

Calculate the value of \(g(x)\)

So, when substituting \(x=2\) in the equation, we get \(g(2) = \frac{1}{3} \cdot (2)^2 = \frac{4}{3}\). So, when \(x\) is \(2\), \(g(x)\) is \(\frac{4}{3}\).

Key concepts

Parabola Graph

When you hear the term 'quadratic function', think of a 'U'-shaped curve on a graph, which is called a parabola. Every parabola has distinct features, such as a vertex—the highest or lowest point, an axis of symmetry, and a direction in which it opens, either upward or downward. The general form of a quadratic function is \( y=ax^2+bx+c \), where \( a \), \( b \), and \( c \) are constants. The graph of the quadratic function mentioned in the exercise, \( g(x)=\frac{1}{3}x^2 \), represents a parabola that opens upwards, since the coefficient of \( x^2 \) is positive.

When graphing a parabola, it's important to plot several points by choosing different values for \( x \) and computing the corresponding \( y \) or \( g(x) \) values. Next, mark these points on the coordinate plane and draw a smooth curve to see the shape of the parabola. By studying this graph, you can understand the behavior of the quadratic function across different values of \( x \) and predict the outcomes for \( g(x) \) based on its shape and position.

Coefficient of Quadratic Term

In a quadratic function, such as \( y=ax^2+bx+c \), the coefficient of the quadratic term, represented by \( a \) in the equation, plays a crucial role in determining the parabola's shape and direction. This coefficient affects how 'wide' or 'narrow' the parabola is and whether it opens upwards or downwards. An upward opening parabola occurs when the coefficient is positive, and a downward opening parabola occurs when it's negative.

For instance, in the function \( g(x)=\frac{1}{3}x^2 \), the coefficient \( \frac{1}{3} \) means the parabola opens upward and will be wider than the parabola of \( f(x)=x^2 \), because the absolute value of \( \frac{1}{3} \) is less than one. If the absolute value of the coefficient is greater than one, the parabola would be narrower. Therefore, this coefficient shapes the parabola and influences how quickly the function's values increase or decrease as \( x \) moves away from the vertex.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. A quadratic equation can typically be written in the standard form of \( ax^2+bx+c=0 \). There are various methods to solve these equations, such as factoring, completing the square, using the quadratic formula, and graphing. Each method has its uses depending on the form of the equation and the values of \( a \), \( b \), and \( c \).

For our example equation, \( y=\frac{1}{3}x^2 \), if we were setting it to zero to find the roots of the equation (the points where the graph crosses the \( x \) - axis), we would get \( \frac{1}{3}x^2=0 \). In this simple case, \( x \) is zero. Generally, when equations are more complex, you might need to apply one of the aforementioned methods. The roots or solutions of the equation are crucial because they provide the \( x \) - intercepts of the parabola, and they are the values of \( x \) that make \( g(x) \) equal zero. Understanding how to find these solutions enables you to analyze the function fully and to predict and calculate key features of its graph.