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Chapter 2: Problem 54

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$g(x)=x^{2}-1$$

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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Most popular questions from this chapter

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