Problem 62
Question
Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$g(x)=\frac{1}{2}(x-1)^{2}$$
Step-by-Step Solution
Verified Answer
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).
1Step 1: 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.
2Step 2: 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.
3Step 3: 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.
Other exercises in this chapter
Problem 62
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