Problem 53 - Precalculus Essentials

Question

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$g(x)=x^{2}-2$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = x^{2} - 2\) is a parabola opening upwards with its vertex at the point (0, -2). This is obtained by shifting the graph of the standard quadratic function \(f(x) = x^{2}\) down by 2 units.

Step 1: Graph the Parent Function

The parent function here is \(f(x) = x^{2}\), which is a standard quadratic function. Its graph will be a parabola opening upwards with its vertex at the origin (0, 0). This is because, for each x-value, \(-x^{2}\) and \(x^{2}\) are the same, thus making 'x=0' the axis of symmetry.

Step 2: Identify the Transformation

In the function \(g(x) = x^{2} - 2\), compared to the parent function \(f(x) = x^{2}\), we subtract 2 from each output (or y-value). This is a vertical shift or translation that moves the graph down by 2 units.

Step 3: Graph the Transformed Function

To graph \(g(x)\), we take the graph of \(f(x)\) and shift it down by 2 units. The vertex of the graph will now be at (0, -2), i.e., it will move from the origin to the point (0, -2). The shape of the parabola and the direction it opens remain the same, only its position changes.