Problem 54 - 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. $$g(x)=x^{2}-1$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = x^{2} - 1\) is a parabola that opens upwards with its vertex at the point (0,-1).

Step 1: Graphing the Standard Quadratic Function

Start by graphing the standard quadratic function, \(f(x)=x^{2}\). This is a parabola that opens upwards with its vertex at the origin (0,0). It is symmetric about the y-axis, and as x moves away from the origin, \(f(x)\) increases.

Step 2: Identifying the Transformation

Next, identify the transformation that needs to be applied to the standard graph to obtain the function \(g(x) = x^{2} - 1\). Here, the value under \(x^{2}\) subtracts 1, indicating a vertical shift. Specifically, it's a downward shift since we are subtracting from \(x^{2}\). This shift moves every point on \(f(x)=x^{2}\), one unit down to graph \(g(x) = x^{2} - 1\).

Step 3: Applying the Transformation and Graphing the Function

Apply the transformation to the graph of \(f(x)=x^{2}\). You will get the graph of \(g(x) = x^{2} - 1\) by moving every point on the graph of \(f(x)=x^{2}\) one unit downwards.