Question

directions
step 1: draw/trace any logo or picture on graph 1.
step 2: on graph 1, pick at least 8 points to label with letters a, b, c, and so on.
step 3: choose a scale factor to dilate by. (choose ½, 2 or 3)
step 4: fill out the table with the pre - image and image coordinates.
step 5: draw the dilated image on graph 2.
step 6: answer the analysis questions.

pre - image coordinates
a ( , )
b ( , )
c ( , )
d ( , )
e ( , )
f ( , )
g ( , )
h ( , )
i ( , )
j ( , )
( , )
( , )
( , )
( , )
( , )

image coordinates
a ( , )
b ( , )
c ( , )
d ( , )
e ( , )
f ( , )
g ( , )
h ( , )
i ( , )
j ( , )
k ( , )
l ( , )
m ( , )
n ( , )

my scale factor: r =

analysis questions

1. imagine a student was given the dilation that you made. would they be able to use the coordinates to find the scale factor? explain.

1. there are 2 pieces of information you can use to determine whether a dilation is a reduction or an enlargement. what type of dilation did you make, and what are the 2 ways a student could determine the dilation you made?

Explanation:

Step1: Choose Scale Factor

Let's choose a scale factor \( r = 2 \). This means we will multiply each coordinate of the pre - image points by 2 to get the image points.

Step2: Label Pre - Image Points

Looking at Graph 1 (the apple - shaped figure), we pick 8 points:

• Point A: Let's say \( A=(2,3) \)
• Point B: \( B=(7,3) \)
• Point C: \( C=(7,7) \)
• Point D: \( D=(3,7) \)
• Point E: \( E=(5,3) \)
• Point F: \( F=(7,5) \)
• Point G: \( G=(5,7) \)
• Point H: \( H=(3,5) \)

Step3: Calculate Image Points (Dilation)

For a dilation with scale factor \( r = 2 \), if a pre - image point is \( (x,y) \), the image point \( (x',y') \) is given by \( x'=r\times x \) and \( y'=r\times y \)

• For \( A=(2,3) \): \( A'=(2\times2,3\times2)=(4,6) \)
• For \( B=(7,3) \): \( B'=(7\times2,3\times2)=(14,6) \)
• For \( C=(7,7) \): \( C'=(7\times2,7\times2)=(14,14) \)
• For \( D=(3,7) \): \( D'=(3\times2,7\times2)=(6,14) \)
• For \( E=(5,3) \): \( E'=(5\times2,3\times2)=(10,6) \)
• For \( F=(7,5) \): \( F'=(7\times2,5\times2)=(14,10) \)
• For \( G=(5,7) \): \( G'=(5\times2,7\times2)=(10,14) \)
• For \( H=(3,5) \): \( H'=(3\times2,5\times2)=(6,10) \)

Step4: Fill the Table

Pre - Image Coordinates	Image Coordinates
\( B(7,3) \)	\( B'(14,6) \)
\( C(7,7) \)	\( C'(14,14) \)
\( D(3,7) \)	\( D'(6,14) \)
\( E(5,3) \)	\( E'(10,6) \)
\( F(7,5) \)	\( F'(14,10) \)
\( G(5,7) \)	\( G'(10,14) \)
\( H(3,5) \)	\( H'(6,10) \)

Step5: Answer Analysis Questions

Question 1

Yes, a student can find the scale factor using the coordinates. The formula for dilation is \( (x',y')=(r\times x,r\times y) \), so \( r=\frac{x'}{x}=\frac{y'}{y} \) (as long as \( x
eq0 \) and \( y
eq0 \)). For example, for point \( A=(2,3) \) and \( A'=(4,6) \), \( r = \frac{4}{2}=\frac{6}{3}=2 \). By taking the ratio of the \( x \) - coordinates (or \( y \) - coordinates) of the image and pre - image points, the student can find the scale factor.

Question 2

The two ways to determine if a dilation is a reduction or an enlargement are:

1. Compare the scale factor to 1: If \( r>1 \), it is an enlargement; if \( 0 < r<1 \), it is a reduction. Since our scale factor \( r = 2>1 \), the dilation is an enlargement.
2. Compare the size of the image to the pre - image: If the image is larger than the pre - image, it is an enlargement; if it is smaller, it is a reduction. Looking at the coordinates, the image points have larger \( x \) and \( y \) values than the pre - image points (e.g., \( A=(2,3) \) and \( A'=(4,6) \)), so the image is larger, confirming it is an enlargement.

Answer:

Table Filling (Example with \( r = 2 \)):

Pre - Image Coordinates	Image Coordinates
\( B(7,3) \)	\( B'(14,6) \)
\( C(7,7) \)	\( C'(14,14) \)
\( D(3,7) \)	\( D'(6,14) \)
\( E(5,3) \)	\( E'(10,6) \)
\( F(7,5) \)	\( F'(14,10) \)
\( G(5,7) \)	\( G'(10,14) \)
\( H(3,5) \)	\( H'(6,10) \)

Analysis Question 1 Answer:

Yes. Using \( r=\frac{x'}{x}=\frac{y'}{y} \) (for non - zero \( x,y \)), e.g., \( A(2,3)\to A'(4,6) \), \( r = \frac{4}{2}=\frac{6}{3}=2 \).

Analysis Question 2 Answer:

Type: Enlargement. Ways: 1. \( r = 2>1 \) (scale factor > 1). 2. Image size > Pre - image size (coordinates show larger values).