Question

the pre-image ( abcd ) is dilated to become the image ( abcd ).
the origin is the center of dilation.
what scale factor is used to create the dilation?
graph with ( y )-axis (0–10), ( x )-axis (0–10), red points ( a, b, c, d ), blue points ( a, b, c, d ), and grid lines
options:
( circ 2 )
( circ -2 )
( circ -\frac{1}{2} )
( circ \frac{1}{2} )

Explanation:

Step1: Find coordinates of a point and its image

Let's take point \( A \) and its image \( A' \). From the graph, \( A \) has coordinates \( (1, 4) \) and \( A' \) has coordinates \( (2, 8) \).

Step2: Calculate the scale factor

The scale factor \( k \) for dilation centered at the origin is given by the ratio of the coordinates of the image to the pre - image. For the \( x \) - coordinate: \( k=\frac{x_{A'}}{x_{A}}=\frac{2}{1} = 2 \). For the \( y \) - coordinate: \( k=\frac{y_{A'}}{y_{A}}=\frac{8}{4}=2 \). We can also check with another point, say \( B(3,4) \) and \( B'(6,8) \). \( \frac{6}{3}=2 \) and \( \frac{8}{4} = 2 \). Since the image is larger than the pre - image and the dilation is a enlargement (positive scale factor as the direction from pre - image to image is the same from the origin), the scale factor is 2.

Answer:

2