Question

1. the line whose equation is 6x + 3y = 3 is dilated by a scale factor of 2 centered at the point (0,0). an equation of its image is

a. y = -2x + 1
b. y = -2x + 2
c. y = -4x + 1
d. y = -4x + 2

1. triangle abc is the image of △abc after a dilation centered at the origin. the coordinates of the vertices of △abc are a(-2,1), b(2,4), and c(2,-3).

image of a coordinate grid with triangle abc
if the coordinates of a are (-4,2), the coordinates of b are
a. (8,4)
b. (4,8)
c. (4,-6)
d. (1,2)

1. which transformation does not always preserve distance?

a. (x,y) → (x + 2, y)
b. (x,y) → (-y, -x)
c. (x,y) → (2x, y - 1)
d. (x,y) → (3 - x, 2 - y)

Explanation:

Question 4

Step1: Rewrite original line equation

Rewrite \(6x + 3y = 3\) in slope - intercept form (\(y=mx + b\)). Subtract \(6x\) from both sides: \(3y=-6x + 3\). Divide by 3: \(y=-2x + 1\). The slope \(m=-2\), y - intercept \(b = 1\).

Step2: Analyze dilation of a line

When a line is dilated with a scale factor \(k\) centered at the origin, if the line does not pass through the origin, the image of the line is parallel to the original line (same slope) and we can find the new y - intercept by considering a point on the original line. Let's take the y - intercept point \((0,1)\) on \(y=-2x + 1\). After dilation with scale factor 2 centered at \((0,0)\), the point \((0,1)\) becomes \((0\times2,1\times2)=(0,2)\).
Since the slope of the line remains the same (\(m=-2\)) after dilation (because dilation of a line not through the origin gives a parallel line), the equation of the image line is \(y=-2x+2\).

Step1: Find the scale factor of dilation

We know that the original coordinate of \(A\) is \((-2,1)\) and the image coordinate \(A'\) is \((-4,2)\). To find the scale factor \(k\), we use the formula for dilation centered at the origin: \((x',y')=(k\times x,k\times y)\). For the \(x\) - coordinate: \(-4=k\times(-2)\), so \(k = \frac{-4}{-2}=2\). For the \(y\) - coordinate: \(2=k\times1\), so \(k = 2\).

Step2: Apply the scale factor to point \(B\)

The original coordinate of \(B\) is \((2,4)\). Using the scale factor \(k = 2\) for dilation centered at the origin, the coordinates of \(B'\) are \((k\times2,k\times4)=(2\times2,2\times4)=(4,8)\).

Step1: Analyze option A

The transformation \((x,y)\to(x + 2,y)\) is a translation (horizontal shift). Translations preserve distance because all points are moved by the same amount in the same direction.

Step2: Analyze option B

The transformation \((x,y)\to(-y,-x)\) is a rotation (90 - degree rotation about the origin followed by a reflection, or a single rotation/reflection). Rotations and reflections preserve distance.

Step3: Analyze option C

The transformation \((x,y)\to(2x,y - 1)\) is a non - uniform dilation (scale factor 2 in the \(x\) - direction) and a translation (shift down by 1 unit in the \(y\) - direction). A non - uniform dilation (scaling in one direction) does not preserve distance. For example, if we have two points \((0,0)\) and \((1,0)\), the distance between them is \(1\). After transformation, \((0,0)\to(0,-1)\) and \((1,0)\to(2,-1)\). The distance between \((0,-1)\) and \((2,-1)\) is \(2\), which is different from the original distance.

Step4: Analyze option D

The transformation \((x,y)\to(3 - x,2 - y)\) is a combination of reflections (reflection over the line \(x=\frac{3}{2}\) and \(y = 1\)). Reflections preserve distance.

Answer:

B. \(y = - 2x+2\)

Question 5