Problem 53 - College Algebra 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 function \(g(x) = x^{2}-2\) is a downward translation by 2 units of the graph of the standard quadratic function \(f(x) = x^{2}\). The graph is a parabola that opens upwards with its vertex at the point (0, -2).

Step 1: Graph the standard quadratic function

The standard quadratic function is \(f(x)=x^{2}\). It is a parabola that opens upwards with vertex at (0,0). The graph intersects the y-axis at the origin and the x-axis at the origin.

Step 2: Understand the transformation

By comparing \(f(x) = x^{2}\) and \(g(x) = x^{2}-2\), it can be seen that there is a transformation, specifically a vertical shift downwards by a factor of 2. This means each y-coordinate from the original graph will be reduced by 2 to obtain the graph of \(g(x)\).

Step 3: Apply the transformation and graph \(g(x)\)

To graph \(g(x) = x^{2}-2\), keep the x-coordinates the same as in the graph of \(f(x)\), but subtract 2 from each of the y-coordinates. In doing so, the original vertex (0,0) will move to (0,-2) and the graph will be a downward shift of the parabola \(y=x^{2}\) by 2 units. This means the new parabola still opens upwards but now its vertex is at (0,-2) and it intersects the y-axis at (0,-2).