Question

The line given by 2, y, equals, 4, x, minus, 22y=4x−2 is dilated by a scale factor centered at the origin. The image of the line after dilation is given by minus, 4, x, plus, 2, y, equals, minus, 6−4x+2y=−6. What is the scale factor of the dilation?

Answer 1

Answer:

  3

Step-by-step explanation:

You want the dilation scale factor that moves the line 2y = 4x -2 to the line -4x +2y = -6.

Form

Writing the equations in the same form, we have ...

• original line: 2y = 4x -2
• dilated line: 2y = 4x -6 . . . . . . . add 4x to the given equation

Factor

We notice that the constant changed by a factor of 3 from -2 to -6. This is the dilation factor.

The dilation factor is 3.

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Additional comment

When the coefficients of the variables are the same, the constant in the equation can be thought of as representing the distance of the line from the origin. Multiplying the distance by 3 multiplies that constant by 3.

The actual distance from the origin is 2/√(2²+4²) = 2/√20 = 1/√5 prior to dilation. After dilation, it is 6/√20 = 3/√5, a factor of 3 farther.

In the attached graph, you will notice the x- and y-intercepts are 3 times as far from the origin for the dilated line.