Question

1. graph the image of the rectangle below using a scale factor of k = 3. 16. graph the image of the triangle below using a scale factor of k = 1/2. 17. graph the image of the quadrilateral below using a scale factor of k = 3/2. 18. graph the image of the triangle below using a scale factor of k = 3/4. 19. identify the scale factor used to graph the image below. 20. identify the scale factor used to graph the image below.

Explanation:

Step1: Recall dilation formula

If a point $(x,y)$ is dilated with a scale - factor $k$ centered at the origin, the new point $(x',y')$ is given by $(x',y')=(k\cdot x,k\cdot y)$.

Step2: For problem 15 with $k = 3$

Given $K(-2,3)$, $L(2,1)$, $M(1, - 1)$, $N(-3,1)$
$K'=(3\times(-2),3\times3)=(-6,9)$
$L'=(3\times2,3\times1)=(6,3)$
$M'=(3\times1,3\times(-1))=(3,-3)$
$N'=(3\times(-3),3\times1)=(-9,3)$

Step3: For problem 16 with $k=\frac{1}{2}$

Assume the vertices of the triangle are $B$, $C$, $D$ with coordinates (not given in the problem - statement clearly, but using the formula)
If a vertex has coordinates $(x,y)$, the new coordinates are $(\frac{1}{2}x,\frac{1}{2}y)$

Step4: For problem 17 with $k = \frac{3}{2}$

For a point $(x,y)$ of the quadrilateral, the new point is $(\frac{3}{2}x,\frac{3}{2}y)$

Step5: For problem 18 with $k=\frac{3}{4}$

For a point $(x,y)$ of the triangle, the new point is $(\frac{3}{4}x,\frac{3}{4}y)$

Step6: For problem 19 and 20

To find the scale - factor $k$, if we know a pre - image point $(x,y)$ and its image point $(x',y')$, we use the formula $k=\frac{x'}{x}=\frac{y'}{y}$ (assuming the center of dilation is the origin)

Answer:

For problem 15: $K'(-6,9)$, $L'(6,3)$, $M'(3,-3)$, $N'(-9,3)$
For problem 16: Use $(x',y') = (\frac{1}{2}x,\frac{1}{2}y)$ for each vertex
For problem 17: Use $(x',y')=(\frac{3}{2}x,\frac{3}{2}y)$ for each vertex
For problem 18: Use $(x',y') = (\frac{3}{4}x,\frac{3}{4}y)$ for each vertex
For problem 19 and 20: Use $k=\frac{x'}{x}=\frac{y'}{y}$ (center of dilation at origin) to find $k$ for given pre - image and image points.