Question

rose graphs this equation as shown. (x^{2}-4x + y^{2}=0). rose correctly found the center but incorrectly found the radius. to correct her graph, she should: increase the radius to 16; shift the center to (2,1); decrease the radius to 2; shift the center to (-2,0)

Explanation:

Step1: Rewrite the circle - equation in standard form

The general equation of a circle is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center and \(r\) is the radius. Given \(x^{2}-4x + y^{2}=0\), complete the square for the \(x\) - terms. \(x^{2}-4x=(x - 2)^2-4\). So the equation becomes \((x - 2)^2-4 + y^{2}=0\), which can be rewritten as \((x - 2)^2+y^{2}=4\).

Step2: Identify the center and radius

From the standard - form equation \((x - 2)^2+y^{2}=4\), the center of the circle is \((2,0)\) and the radius \(r = 2\). If the center is correctly found but the radius is incorrect, and the current radius in the graph seems to be larger than the actual radius. To correct the graph, we need to decrease the radius to 2.

Answer:

decrease the radius to 2