Question

Please helppp, I'm using all my points!! Line A passes through the points (-3,-4) and (-6,-5). Write the equation of the image of A after a dilation with a scale factor of 2. Write your in answer in slope-intercept form.

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Answer 1

Answer:

The equation of the image of A after a dilation with a scale factor of 2 will be:

• $y\:=\:\frac{1}{3}x\:-6$

Step-by-step explanation:

Some background Knowledge about dilation.

We know that when an object is dilated by a scale factor, it gets reduced, stretched, or remains the same, depending upon the value of the scale factor.

• If the scale factor > 1, the image is enlarged

• If the scale factor is between 0 and 1, it gets shrunk

• If the scale factor = 1, the object and the image are congruent

Rule to calculate the dilation by a scale factor 2 centered at the origin

P(x, y) → P'(2x, 2y)

Here, P'(2x, 2y) is the image of P(x, y).

• It means the coordinates of the image can be determined by multiplying the coordinates of the original point by 2.

Now, let us solve our case:

Given the points

• (-3, -4)

• (-6, -5)

Let say the points are X(-3, -4) and Y(-6, -5)

Rule to calculate the dilation by a scale factor 2 centered at the origin

P(x, y) → P'(2x, 2y)

so

X(-3, -4)    →   X' (2(-3), 2(-4))    →    X'(-6, -8)

Y(-6, -5)    →   Y' (2(-6), 2(-5))    →    Y'(-12, -10)

Thus, after the dilation scale of factor 2, now the image line will through the points X'(-6, -8) and Y'(-12, -10)

so we have the points of the image line

• X'(-6, -8)

• Y'(-12, -10)

Determining the slope between X'(-6, -8) and Y'(-12, -10)

$\left(x_1,\:y_1\right)=\left(-6,\:-8\right),\:\left(x_2,\:y_2\right)=\left(-12,\:-10\right)$

$m=\frac{-10-\left(-8\right)}{-12-\left(-6\right)}$

$m=\frac{1}{3}$

The slope-intercept form of the line equation

y = mx+b

where

• m is the slope

• b is the y-intercept

now substituting m = 1/3 and (-6, -8) in the slope-intercept form of the line equation

$y = mx + b$

$-8\:=\:\frac{1}{3}\left(-6\right)\:+\:b$

$-2+b=-8$

Add 3 to both sides

$-2+b+2=-8+2$

Simplify

$b=-6$

now substituting b = -6 and m = 1/3 in the slope-intercept form of line equation

$y = mx + b$

$y\:=\:\frac{1}{3}x\:+\:\left(-6\right)$

$y\:=\:\frac{1}{3}x\:-6$

Therefore, the equation of the image of A after a dilation with a scale factor of 2 will be:

• $y\:=\:\frac{1}{3}x\:-6$