The general formula for a circle with center (h,k) and radius r is \(\displaystyle (x-h)^2+(y-k)^2=r^2\).

The center of the original circle, therefore, is (2, -4). 

If we move the circle to the left 2 spaces and down 3 spaces, then the center of the new circle is given by \(\displaystyle (2-2, -4-3)\) or \(\displaystyle (0,-7)\). 

In order to reflect a function across the y-axis, all of the x-coordinates of every point on that function must be multiplied by negative one. However, the y-values of each point on the function will not change. Thus, we can represent the reflection of f(x) across the y-axis as f(-x). The figure below shows a generic function (not f(x) given in the problem) that has been reflected across the y-axis, in order to offer a better visual understanding. 

Therefore, g(x) = f(–x).

f(x) = x³ – 2x² + x – 4

g(x) = f(–x) = (–x)³ – 2(–x)² + (–x) + 4

g(x) = (–1)³x³ –2(–1)²x² – x + 4

g(x) = –x³ –2x² –x + 4.

The answer is –x³ –2x² –x + 4.

If Jenny moves Bobby's circle up 2 units and to the right 1 unit, then the center of her circle is (3, 7). The radius remains 10.

The general equation for a circle with center (h, k) and radius r is given by

\(\displaystyle (x-h)^2+(y-k)^2=r^2\)

For Jenny's circle, (h, k) = (3, 7) and r=10.

Substituting these values into the general equation gives us

\(\displaystyle (x-3)^2+(y-7)^2=100\)

A plane can be named after any three points on the plane that are not on the same line. As seen below, points \(\displaystyle C\), \(\displaystyle E\),  and \(\displaystyle F\) are on the same line. 

Therefore, Plane \(\displaystyle CEF\) is not a valid name for the plane.

Two rays are opposite rays, by definition, if 

(1) they have the same endpoint, and

(2) their union is a line.

The first letter in the name of a ray refers to its endpoint; the second refers to the name of any other point on the ray. \(\displaystyle \overrightarrow{EC}\) and \(\displaystyle \overrightarrow{EF}\) both have endpoint \(\displaystyle E\), so the first criterion is met. \(\displaystyle \overrightarrow{EC}\) passes through point \(\displaystyle C\) and \(\displaystyle \overrightarrow{EF}\) passes through point \(\displaystyle F\); \(\displaystyle \overrightarrow{EC}\) and \(\displaystyle \overrightarrow{EF}\) are indicated below in green and red, respectively:

The union of the two rays is a line. Both criteria are met, so the rays are indeed opposite.

A line can be named after any two points it passes through. The line \(\displaystyle \overleftrightarrow{CF}\) is indicated in green below.

The line does not pass through \(\displaystyle D\), so \(\displaystyle D\) cannot be part of the name of the line. Specifically, \(\displaystyle \overleftrightarrow{DF}\) is not a valid name.

By definition, two angles comprise a pair of vertical angles if 

(1) they have the same vertex; and

(2) the union of the two angles is exactly a pair of intersecting lines.

In the figure below, \(\displaystyle \angle 1\) and \(\displaystyle \angle 2\) are marked in green and red, respectively:

While the two angles have the same vertex, their union is not a pair of intersecting lines. The two angles are not a vertical pair.

By definition, two angles form a linear pair if and only if 

(1) they have the same vertex;

(2) they share a side; and,

(3) their interiors have no points in common.

In the figure below, \(\displaystyle \angle 1\) and \(\displaystyle \angle 2\) are marked in green and red, respectively:

The two angles have the same vertex and share no interior points. However, they do not share a side. Therefore, they do not comprise a linear pair.