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Chapter 2: Problem 62

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

Short Answer

Expert verified
The function \(g(x) = \frac{1}{2}(x - 1)^2\) is a transformation of \(f(x) = x^2\), which involves a shift to the right by one unit and a vertical compression by a factor of 1/2. The graph of \(g(x)\) is a parabola that opens upwards, with the vertex at the point (1,0).

Step by step solution

01

Graph the standard quadratic function

The standard quadratic function is \(f(x) = x^2\). It is a parabola that opens upwards, with the vertex at the origin (0,0). The parabola is symmetric about the y-axis.
02

Apply the horizontal shift

The function \(g(x) = \frac{1}{2}(x - 1)^2\) has a horizontal shift to the right by 1 unit. This is denoted by the '(x - 1)' in the function. This shift moves every point on the graph of \(f(x) = x^2\) one unit to the right.
03

Apply the vertical compression

The function \(g(x) = \frac{1}{2}(x - 1)^2\) also has a vertical compression by a factor of 1/2. This is denoted by the '\(\frac{1}{2}\)' multiplier in the function. Every y-coordinate on the \(f(x) = x^2\) is halved to produce the graph of the given function.

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Most popular questions from this chapter

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