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Translating to a Line Through the Origin

Author:I-Heng McComb
Any line through the origin can be represented by an equation of the form y = mx where m is the slope of the line. We can use this form together with translation to write an equation for any line where we know the slope and the coordinates of one point (otherwise known as point-slope form).

Choosing a Translation

Add a translation vector to diagram which will map Point A to the origin. What is the translation vector in component notation?

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Algebraic Description

We can describe the translation as (x,y) --> (x',y'). Give algebraic expressions for x'and y' in terms of x and y.

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Image of the Line

Now use your translation vector to translate Line AB. The result should be a line through the origin. We can write an equation y' = ___ x' for Line A'B', where (x',y') represents any point on the image line. What is the equation in this case?

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Equation of the Line

What equation do you get when you write your equation for Line A'B' in terms of x and y (the coordinates before you translated)?

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Slope-Intercept Form Equation

Rewrite your equation for Line AB in slope-intercept (y = mx + b) form. (Does the equation you found have the correct y-intercept? Move the graphic view to find out!)

Further Exploration:

1. Repeat the above process using Point B instead of Point A. Compare the results with what you did above. 2. Want more practice? Change Line AB by moving Points A and B and try again!

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