A dilation is a transformation that produces an image that is the same shape as the original but is a different size. Given a certain scale factor, the coordinates of each point in the image are a fixed multiple of the coordinates of the corresponding point in the original figure.
In this case, we need to demonstrate a dilation centered at the origin with a scale factor of 1.5.
Let's start with the coordinates given for a point [tex]\((x, y)\)[/tex]. The formula for dilating this point is given by:
[tex]\[
(x, y) \rightarrow (1.5 \cdot x, 1.5 \cdot y)
\][/tex]
We need to show that for any coordinate [tex]\((x, y)\)[/tex], the dilated point [tex]\((x_dilated, y_dilated)\)[/tex] follows this transformation.
Let's consider a specific point [tex]\((1, 1)\)[/tex] to illustrate this dilation.
1. Identify the original coordinates:
[tex]\[
(x, y) = (1, 1)
\][/tex]
2. Apply the dilation with scale factor 1.5:
[tex]\[
x_dilated = 1.5 \cdot x = 1.5 \cdot 1 = 1.5
\][/tex]
[tex]\[
y_dilated = 1.5 \cdot y = 1.5 \cdot 1 = 1.5
\][/tex]
3. Determine the new coordinates after dilation:
[tex]\[
(x_dilated, y_dilated) = (1.5, 1.5)
\][/tex]
So, the point [tex]\((1, 1)\)[/tex] after dilation becomes [tex]\((1.5, 1.5)\)[/tex].
The scale factor 1.5 affects both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates equally, which means every coordinate [tex]\((x, y)\)[/tex] will be mapped to [tex]\((1.5x, 1.5y)\)[/tex] by this transformation.
Thus, we can now confidently choose the transformation [tex]\((x, y) \rightarrow (1.5x, 1.5y)\)[/tex] as a demonstration of performing a dilation centered at the origin with the given scale factor 1.5. The resulting points illustrate that both coordinates are scaled by the factor uniformly, confirming the rules of dilation.
