graph the image of rectangle cdef after a dilation with a scale factor of 2, centered at the origin.

Answer

Explanation:

Step1: Identify coordinates

First, find the coordinates of the vertices of rectangle ( CDEF ). From the graph:

( C(-4, -5) ) (Wait, looking at the graph, ( C ) is at ( (-4, -5) )? Wait, no, the grid: ( F ) is at ( (-4, 1) )? Wait, no, the y - axis: the blue points ( F ) is at ( (-4, 1) )? Wait, no, the graph shows ( F ) at ( (-4, 1) )? Wait, no, the y - coordinate for ( F ) and ( E ) is 1? Wait, the grid lines: the horizontal line for ( F ) and ( E ) is at ( y = 1 )? Wait, no, the original graph: ( F ) is at ( (-4, 1) )? Wait, no, looking at the y - axis, the numbers are 10, 8, 6, 4, 2, 0, - 2, - 4, - 6. So ( F ) is at ( (-4, 1) )? Wait, no, the blue dot for ( F ) is on the line ( y = 1 ) (since it's between ( y = 0 ) and ( y = 2 ), at ( y = 1 )) and ( x=-4 ). ( E ) is at ( (5, 1) )? Wait, no, the x - coordinate for ( E ) is 5? Wait, the grid: ( E ) is at ( (5, 1) )? Wait, no, the x - axis labels: - 10, - 8, - 6, - 4, - 2, 0, 2, 4, 5? No, the x - axis has marks at - 10, - 8, - 6, - 4, - 2, 0, 2, 4, 5? No, the x - axis is from - 10 to 10, with grid lines every 1 unit. Wait, ( F ) is at ( (-4, 1) ), ( E ) is at ( (5, 1) )? No, the x - coordinate for ( E ) is 5? Wait, no, the blue dot for ( E ) is at ( x = 5 )? Wait, no, the original graph: ( F(-4,1) ), ( E(5,1) ), ( D(5, - 5) ), ( C(-4, - 5) )? Wait, no, ( D ) is at ( (5, - 5) ) and ( C(-4, - 5) ). Wait, let's re - examine:

The rectangle ( CDEF ) has vertices: ( F(-4,1) ), ( E(5,1) ), ( D(5, - 5) ), ( C(-4, - 5) ). Wait, no, the vertical lines for ( C ) and ( F ) are at ( x=-4 ), and for ( D ) and ( E ) at ( x = 5 ). The horizontal lines for ( F ) and ( E ) at ( y = 1 ), and for ( C ) and ( D ) at ( y=-5 ).

Dilation with scale factor ( k = 2 ) centered at the origin. The rule for dilation centered at the origin is ( (x,y)\to(kx,ky) ).

So for each vertex:

Vertex ( F(-4,1) ): After dilation, ( F'=(-4\times2,1\times2)=(-8,2) )
Vertex ( E(5,1) ): After dilation, ( E'=(5\times2,1\times2)=(10,2) )
Vertex ( D(5, - 5) ): After dilation, ( D'=(5\times2,-5\times2)=(10, - 10) )
Vertex ( C(-4, - 5) ): After dilation, ( C'=(-4\times2,-5\times2)=(-8, - 10) )

Wait, maybe I made a mistake in the original coordinates. Let's look again:

Looking at the graph, ( F ) is at ( (-4,1) ), ( E ) is at ( (5,1) ), ( D ) is at ( (5, - 5) ), ( C ) is at ( (-4, - 5) ).

Dilation with scale factor 2 centered at the origin:

For a point ( (x,y) ), the image after dilation ( (x',y')=(2x,2y) )

So:

( F(-4,1)\to F'(-8,2) )
( E(5,1)\to E'(10,2) )
( D(5, - 5)\to D'(10, - 10) )
( C(-4, - 5)\to C'(-8, - 10) )

Now, to graph the image:

Plot the points ( F'(-8,2) ), ( E'(10,2) ), ( D'(10, - 10) ), ( C'(-8, - 10) ) and connect them to form the dilated rectangle.

Step1: Find original coordinates

First, identify the coordinates of the vertices of rectangle ( CDEF ). From the graph:

( F(-4,1) )
( E(5,1) )
( D(5, - 5) )
( C(-4, - 5) )

Step2: Apply dilation rule

The rule for dilation with scale factor ( k = 2 ) centered at the origin is ( (x,y)\to(2x,2y) ).

For ( F(-4,1) ): ( x'=2\times(-4)=-8 ), ( y'=2\times1 = 2 ), so ( F'(-8,2) )
For ( E(5,1) ): ( x'=2\times5 = 10 ), ( y'=2\times1=2 ), so ( E'(10,2) )
For ( D(5, - 5) ): ( x'=2\times5 = 10 ), ( y'=2\times(-5)=-10 ), so ( D'(10, - 10) )
For ( C(-4, - 5) ): ( x'=2\times(-4)=-8 ), ( y'=2\times(-5)=-10 ), so ( C'(-8, - 10) )

Step3: Graph the new vertices

Plot the points ( F'(-8,2) ), ( E'(10,2) ), ( D'(10, - 10) ), and ( C'(-8, - 10) ) on the coordinate plane and connect them in order to form the dilated rectangle.

Answer:

The dilated rectangle has vertices ( F'(-8,2) ), ( E'(10,2) ), ( D'(10, - 10) ), and ( C'(-8, - 10) ). To graph it, plot these points and connect them. (Note: Since the problem asks to graph, the key is to find the new coordinates using the dilation rule and plot them. The final answer is the set of new coordinates for the vertices of the dilated rectangle.)