After A Dilation With A Center Of { (0,0)$}$, A Point Was Mapped As { (4,-6) \rightarrow (12, Y)$}$. A Student Determined { Y$}$ To Be -2. Evaluate The Student's Answer.A. The Student Is Correct. B. The Student

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Evaluating the Student's Answer: A Dilation with a Center of (0,0)

Dilation is a transformation that changes the size of a figure. In this case, we are dealing with a dilation with a center of (0,0). This means that the center of dilation is the origin of the coordinate plane. When a point is dilated with a center of (0,0), the distance of the point from the origin is multiplied by a scale factor.

The Given Information

A point was mapped as (4,-6) → (12, y). This means that the point (4,-6) was dilated to the point (12, y) with a center of (0,0).

The Scale Factor

To find the scale factor, we need to find the ratio of the distance of the dilated point from the origin to the distance of the original point from the origin. The distance of the dilated point (12, y) from the origin is √(12^2 + y^2). The distance of the original point (4,-6) from the origin is √(4^2 + (-6)^2) = √(16 + 36) = √52.

The Scale Factor Formula

The scale factor (k) can be found using the formula:

k = (distance of dilated point from origin) / (distance of original point from origin)

Substituting the values, we get:

k = (√(12^2 + y^2)) / (√52)

Simplifying the Scale Factor Formula

To simplify the formula, we can square both sides:

k^2 = (12^2 + y^2) / 52

k^2 = (144 + y^2) / 52

The Relationship Between the Scale Factor and the Coordinates

Since the dilation is with a center of (0,0), the x-coordinate of the dilated point is the x-coordinate of the original point multiplied by the scale factor. Similarly, the y-coordinate of the dilated point is the y-coordinate of the original point multiplied by the scale factor.

The Relationship Formula

The relationship between the coordinates can be expressed as:

x-coordinate of dilated point = x-coordinate of original point × k y-coordinate of dilated point = y-coordinate of original point × k

Substituting the values, we get:

12 = 4 × k y = -6 × k

Solving for the Scale Factor

From the first equation, we can solve for k:

k = 12 / 4 k = 3

Substituting the Scale Factor into the Second Equation

Now that we have the value of k, we can substitute it into the second equation:

y = -6 × 3 y = -18

Evaluating the Student's Answer

The student determined y to be -2. However, based on our calculations, the correct value of y is -18. Therefore, the student's answer is incorrect.

Conclusion

In conclusion, the student's answer of y = -2 is incorrect. The correct value of y is -18, which can be obtained by using the scale factor of 3 and the relationship between the coordinates.
Frequently Asked Questions (FAQs) about Dilation with a Center of (0,0)

Q: What is dilation with a center of (0,0)?

A: Dilation with a center of (0,0) is a transformation that changes the size of a figure by multiplying the distance of the figure from the origin by a scale factor.

Q: How do I find the scale factor for a dilation with a center of (0,0)?

A: To find the scale factor, you need to find the ratio of the distance of the dilated point from the origin to the distance of the original point from the origin. You can use the formula:

k = (distance of dilated point from origin) / (distance of original point from origin)

Q: What is the relationship between the coordinates of the original point and the dilated point?

A: The x-coordinate of the dilated point is the x-coordinate of the original point multiplied by the scale factor. Similarly, the y-coordinate of the dilated point is the y-coordinate of the original point multiplied by the scale factor.

Q: How do I use the scale factor to find the coordinates of the dilated point?

A: You can use the relationship between the coordinates to find the coordinates of the dilated point. For example, if the original point is (x, y) and the scale factor is k, then the coordinates of the dilated point are:

x-coordinate of dilated point = x-coordinate of original point × k y-coordinate of dilated point = y-coordinate of original point × k

Q: What if the scale factor is not a whole number?

A: If the scale factor is not a whole number, you can still use it to find the coordinates of the dilated point. For example, if the scale factor is 2.5, you can multiply the coordinates of the original point by 2.5 to get the coordinates of the dilated point.

Q: Can I use dilation with a center of (0,0) to enlarge or shrink a figure?

A: Yes, you can use dilation with a center of (0,0) to enlarge or shrink a figure. If the scale factor is greater than 1, the figure will be enlarged. If the scale factor is less than 1, the figure will be shrunk.

Q: What are some real-world applications of dilation with a center of (0,0)?

A: Dilation with a center of (0,0) has many real-world applications, such as:

  • Enlarging or shrinking images
  • Creating scale models of buildings or objects
  • Designing graphics or logos
  • Creating special effects in movies or video games

Q: How can I practice dilation with a center of (0,0)?

A: You can practice dilation with a center of (0,0) by:

  • Using online resources or worksheets to practice dilation problems
  • Creating your own dilation problems and solving them
  • Using real-world objects or images to practice dilation
  • Working with a partner or teacher to practice dilation

Q: What are some common mistakes to avoid when working with dilation with a center of (0,0)?

A: Some common mistakes to avoid when working with dilation with a center of (0,0) include:

  • Forgetting to multiply the coordinates of the original point by the scale factor
  • Using the wrong scale factor or coordinates
  • Not checking the units of the scale factor
  • Not considering the direction of the dilation (enlarging or shrinking)

By avoiding these common mistakes, you can ensure that you are working with dilation with a center of (0,0) correctly and accurately.