Example Questions

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Example Question #8 : Transformation

The following is an equation of a circle: If this circle is moved to the left 2 spaces and down 3 spaces, where does the center of the new circle lie?      Explanation:

The general formula for a circle with center (h,k) and radius r is .

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 or Example Question #9 : Transformation

Let f(x) = x3 – 2x2 + x +4. If g(x) is obtained by reflecting f(x) across the y-axis, then which of the following is equal to g(x)?

–x3 + 2x2 – x + 4

x3 + 2x2 + x + 4

x3 – 2x2 – x + 4

–x3 – 2x2 – x – 4

–x3 – 2x2 – x + 4

–x3 – 2x2 – x + 4

Explanation:

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.

Example Question #10 : Transformation

Bobby draws a circle on graph paper with a center at (2, 5) and a radius of 10.

Jenny moves Bobby's circle up 2 units and to the right 1 unit.

What is the equation of Jenny's circle?      Explanation:

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

Substituting these values into the general equation gives us Example Question #5 : How To Find A Ray Refer to the above diagram. The plane containing the above figure can be called Plane .

True

False

False

Explanation:

A plane can be named after any three points on the plane that are not on the same line. As seen below, points  ,  and are on the same line. Therefore, Plane is not a valid name for the plane.

Example Question #1543 : Basic Geometry Refer to the above figure.

True or false: and comprise a pair of opposite rays.

False

True

True

Explanation:

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. and both have endpoint , so the first criterion is met. passes through point and passes through point  and 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.

Example Question #1548 : Basic Geometry Refer to the above diagram:

True or false: may also called .

True

False

False

Explanation:

A line can be named after any two points it passes through. The line is indicated in green below. The line does not pass through , so cannot be part of the name of the line. Specifically, is not a valid name.

Example Question #301 : Coordinate Geometry Refer to the above diagram.

True or false: and comprise a pair of vertical angles.

False

True

False

Explanation:

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, and 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.

Example Question #42 : How To Find An Angle Of A Line Refer to the above diagram.

True or false: and comprise a linear pair.

False

True

False

Explanation:

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, and 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.

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