Question

\(g(x)=\frac{1}{5} x^2-4\)

Short answer

The function is a quadratic function, its vertex is at (0, -4), and it has its axis of symmetry at \(x=0\). It opens upwards.

Step-by-step solution

Identify the function

The function given is \(g(x)=\frac{1}{5}x^2-4\). This is a quadratic function of the form \(f(x)=a x^2 + c\) where \(a=\frac{1}{5}\) and \(c=-4\).

Identify the vertex

For a quadratic function in the form \(f(x)=a x^2 +c\), the vertex of the parabola is \((0, c)\). Therefore, the vertex of the parabola is \((0, -4)\).

Identify the axis of symmetry

For a quadratic function in the form \(f(x)=a x^2 + c\), the axis of symmetry is the vertical line \(x = 0\). So for the given function, the axis of symmetry is \(x=0\).

Determine the direction of opening

If the coefficient of the \(x^2\) term, \(a\) is positive, then the parabola opens upwards and if \(a\) is negative, the parabola opens downwards. Since in the given function, \(a=\frac{1}{5}>0\), the parabola opens upwards.

Key concepts

Vertex of a Parabola

The vertex of a parabola is a key point that represents the minimum or maximum point on the graph of a quadratic function. In the case of a quadratic function written in the form \(f(x)=a x^2 +c\), identifying the vertex is straightforward when there is no \(b\) term, meaning there is no linear \(x\) term. Here, our function is \(g(x) = \frac{1}{5}x^2 - 4\), which conveniently fits this scenario.

For such functions, the vertex is simply located at \((0, c)\). In our exercise, the vertex thus is at \((0, -4)\). This point \((0, -4)\) is crucial since it also represents either the lowest point if the parabola opens upwards, or the highest point if it opens downwards. The vertex is often a point of symmetry as well, making it pivotal in sketching and understanding the graph.

Axis of Symmetry

The axis of symmetry of a parabola is an imaginary vertical line that passes through the vertex. It essentially splits the parabola into two identical halves, making it easier to understand its geometric properties. For our function \(g(x) = \frac{1}{5}x^2 - 4\), we find that this vertical line is \(x = 0\).

The presence of the axis of symmetry allows one to make predictions about points on the parabola. For every point on one side of the axis, there will be a corresponding point that is equidistant on the opposite side. This is why it is a fundamental property of parabolas and helps in quickly sketching the graph. The line \(x = 0\) is particularly significant here because it restores balance to the left and right sides around the vertex.

Direction of Opening

The direction in which a parabola opens is determined by the sign of the coefficient \(a\) in the quadratic function \(f(x)=a x^2 + c\). Understanding this direction is crucial for knowing whether the vertex represents a minimum or maximum value in the context of the graph.

• If \(a > 0\), the parabola opens upwards, resembling a U-shape, and the vertex is the minimum point.
• If \(a < 0\), the parabola opens downwards, resembling an inverted U, and the vertex is the maximum point.

In our function, \(g(x) = \frac{1}{5}x^2 - 4\), \(a\) is \(\frac{1}{5}\) which is positive. Thus, this tells us that the parabola opens upwards, making the vertex \((0, -4)\) the point at which the function takes its minimum value. Recognizing the direction of opening helps in graphing and interpreting the function's behavior over its domain.