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Chapter 1: Problem 53

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

Short Answer

Expert verified
The graph of the function \(g(x) = x^{2} - 2\) is a parabola opening upwards with its vertex at the point (0, -2). This is obtained by shifting the graph of the standard quadratic function \(f(x) = x^{2}\) down by 2 units.

Step by step solution

01

Graph the Parent Function

The parent function here is \(f(x) = x^{2}\), which is a standard quadratic function. Its graph will be a parabola opening upwards with its vertex at the origin (0, 0). This is because, for each x-value, \(-x^{2}\) and \(x^{2}\) are the same, thus making 'x=0' the axis of symmetry.
02

Identify the Transformation

In the function \(g(x) = x^{2} - 2\), compared to the parent function \(f(x) = x^{2}\), we subtract 2 from each output (or y-value). This is a vertical shift or translation that moves the graph down by 2 units.
03

Graph the Transformed Function

To graph \(g(x)\), we take the graph of \(f(x)\) and shift it down by 2 units. The vertex of the graph will now be at (0, -2), i.e., it will move from the origin to the point (0, -2). The shape of the parabola and the direction it opens remain the same, only its position changes.

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Most popular questions from this chapter

A telephone company offers the following plans. Also given are the piecewise functions that model these plans. Use this information to solve. Plan \(A\) \(\cdot \$ 30\) per month buys 120 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 120 \\\30+0.30(t-120) & \text { if } t>120 \end{array}\right. $$ Plan \(B\) \(\cdot \ 40\) per month buys 200 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$ C(t)=\left\\{\begin{array}{ll} 40 & \text { if } 0 \leq t \leq 200 \\\ 40+0.30(t-200) & \text { if } t>200 \end{array}\right. $$ Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.

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