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Chapter 2: 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 function \(g(x) = x^{2}-2\) is a downward translation by 2 units of the graph of the standard quadratic function \(f(x) = x^{2}\). The graph is a parabola that opens upwards with its vertex at the point (0, -2).

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 vertex at (0,0). The graph intersects the y-axis at the origin and the x-axis at the origin.
02

Understand the transformation

By comparing \(f(x) = x^{2}\) and \(g(x) = x^{2}-2\), it can be seen that there is a transformation, specifically a vertical shift downwards by a factor of 2. This means each y-coordinate from the original graph will be reduced by 2 to obtain the graph of \(g(x)\).
03

Apply the transformation and graph \(g(x)\)

To graph \(g(x) = x^{2}-2\), keep the x-coordinates the same as in the graph of \(f(x)\), but subtract 2 from each of the y-coordinates. In doing so, the original vertex (0,0) will move to (0,-2) and the graph will be a downward shift of the parabola \(y=x^{2}\) by 2 units. This means the new parabola still opens upwards but now its vertex is at (0,-2) and it intersects the y-axis at (0,-2).

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Most popular questions from this chapter

Exercises \(98-100\) will help you prepare for the material covered in the first section of the next chapter. In Exercises \(98-99,\) solve each quadratic equation by the method of your choice. $$ 0=-2(x-3)^{2}+8 $$

use a graphing utility to graph each circle whose equation is given. $$ x^{2}+10 x+y^{2}-4 y-20=0 $$

Exercises \(98-100\) will help you prepare for the material covered in the first section of the next chapter. In Exercises \(98-99,\) solve each quadratic equation by the method of your choice. Use the graph of \(f(x)=x^{2}\) to graph \(g(x)=(x+3)^{2}+1\)

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}+3 x+5 y+\frac{9}{4}=0 $$
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