### All Advanced Geometry Resources

## Example Questions

### Example Question #21 : Transformations

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 #1 : Transformation

What is the period of the function?

**Possible Answers:**

2*π*

3*π*

*π*

1

4*π*

**Correct answer:**

4*π*

The period is the time it takes for the graph to complete one cycle.

In this particular case we have a sine curve that starts at 0 and completes one cycle when it reaches .

Therefore, the period is

### Example Question #471 : Advanced Geometry

Explain how the below function translates:

**Possible Answers:**

5 units up, 7 units left

5 units left, 7 units down

5 units right, 7 units down

5 units down, 7 units right

**Correct answer:**

5 units left, 7 units down

When estimating the translations for a quadratic function we must remember what vertex form for a parabola looks like:

In order to end up with:

We must have the below to end up with a positive 5:

This is what gives us the translation left 5 spaces and down 7 spaces.

### Example Question #21 : Transformation

Assume we have a triangle, , with the following vertices:

, , and

If were reflected across the line , what would be the coordinates of the new vertices?

**Possible Answers:**

**Correct answer:**

When we reflect a point across the line, , we swap the x- and y-coordinates; therefore, in each point, we will switch the x and y-coordinates:

becomes ,

becomes , and

becomes .

The correct answer is the following:

The other answer choices are incorrect because we only use the negatives of the coordinate points if we are flipping across either the x- or y-axis.