How to find the equation of a transformed line segment from its endpoints

Expert intro: This kind of question is really about reading a transformation from geometry and turning it into function notation. The midpoint and length clues matter.

Detailed walkthrough

The wording suggests you are looking for a transformation that is perpendicular to a line segment and has double the length.

If a function graph is transformed by a horizontal or vertical scaling, the algebraic form depends on the axis and the direction of the stretch.

For a function written as $g(x)$ in terms of $f(x)$:

• a vertical stretch by factor 2 gives $g(x)=2f(x)$
• a horizontal stretch by factor 2 gives $g(x)=f(x/2)$
• a reflection changes the sign in front of the relevant coordinate

If your segment endpoints are $(-5,5)$ and $(5,2)$, the midpoint is

$\left(\frac{-5+5}{2},\frac{5+2}{2}\right)=(0,3.5).$

That midpoint is useful if the transformation is centered there.

If you mean the graph should be stretched by a factor of 2 around a center line, a typical function form is

$g(x)=f\left(\frac{x}{2}\right)$

for a horizontal stretch, or

$g(x)=2f(x)$

for a vertical stretch.

Because the prompt is a bit unclear, the exact formula depends on whether the stretch is horizontal or vertical and what the original $f(x)$ is.

💡 Pitfall guide

The biggest trap is assuming every “double the length” clue means a horizontal stretch. In function notation, the meaning depends on which direction the graph is being stretched. Also, “perpendicular” may describe a geometric line, not a function transformation.

🔄 Real-world variant

If the question is about the line segment itself rather than a function graph, the slope of the segment from $(-5,5)$ to $(5,2)$ is

$\frac{2-5}{5-(-5)}=-\frac{3}{10}.$

A perpendicular line would have slope $\frac{10}{3}$. If you want a line through the midpoint $(0,3.5)$, its equation is

$y-3.5=\frac{10}{3}(x-0).$

🔍 Related terms

midpoint, horizontal stretch, perpendicular line