# Advanced Geometry : How to find transformation for an analytic geometry equation

## Example Questions

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### Example Question #21 : Transformations

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 #1 : Transformation

What is the period of the function?

2π

3π

π

1

4π

4π

Explanation:

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:

5 units up, 7 units left

5 units left, 7 units down

5 units right, 7 units down

5 units down, 7 units right

5 units left, 7 units down

Explanation:

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?

Explanation:

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.

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