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

Step-by-Step Solution

Answer

The graph of the function \(g(x)=x^{2}-1\) can be obtained by shifting the graph of the standard quadratic function, \(f(x)=x^{2}\), downward by 1 unit.

Step 1: Sketch the Graph of the Basic Quadratic Function

Start by sketching the graph of standard quadratic function \(f(x)=x^{2}\). With this, a U-shape curve is formed known as a parabola. The vertex of this parabola will be at the origin, \( (0,0) \), and it opens upwards. For this graph, each x-value corresponds to a y-value which is the square of the x-value.

Step 2: Understand the Transformation

The transformation that will be needed to go from \(f(x)=x^{2}\) to \(g(x)=x^{2}-1\) is a vertical shift. In particular, the entire graph of \(f(x)=x^{2}\) will be shifted down by 1 unit because of the subtraction by 1 in the function \(g(x)\). This vertical shift does not affect the shape of the graph, only its position.

Step 3: Sketch the Transformed Graph

After understanding the necessary transformation, the graph of \(g(x)\) can be sketched. This process involves shifting every point on the graph of \(f(x)=x^{2}\) down by 1 unit, as dictated by the transformation. The vertex of the parabola, initially at (0,0), will hence be at (0,-1). The graph retains its parabolic shape and will open upwards, just like the graph of the original function.