### All SAT Math Resources

## Example Questions

### 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?

**Possible Answers:**

**Correct answer:**

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) = x^{3} – 2x^{2} + 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)?

**Possible Answers:**

–x^{3} + 2x^{2} – x + 4

x^{3} + 2x^{2} + x + 4

x^{3} – 2x^{2} – x + 4

–x^{3} – 2x^{2} – x – 4

–x^{3} – 2x^{2} – x + 4

**Correct answer:**

–x^{3} – 2x^{2} – x + 4

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) = x^{3} – 2x^{2} + x – 4

g(x) = f(–x) = (–x)^{3} – 2(–x)^{2} + (–x) + 4

g(x) = (–1)^{3}x^{3} –2(–1)^{2}x^{2} – x + 4

g(x) = –x^{3} –2x^{2} –x + 4.

The answer is –x^{3} –2x^{2} –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?

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

True

False

**Correct answer:**

False

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.

**Possible Answers:**

False

True

**Correct answer:**

True

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 .

**Possible Answers:**

True

False

**Correct answer:**

False

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.

**Possible Answers:**

False

True

**Correct answer:**

False

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.

**Possible Answers:**

False

True

**Correct answer:**

False

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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