Asymptotes of Hyperbolas

Algebra 2 · Axiom Academy

LESSON Asymptotes of Hyperbolas Every hyperbola is funneled between two straight guide lines it races toward forever but never touches — and a simple box shows exactly where they are. Start from a hyperbola in standard form. Around its center sits an invisible rectangle — the central box — reaching a to the left and right and b up and down, so its corners land at . set the box's half-width and half-height. Draw the two diagonals of that box through the center and extend them . Those extended diagonals are the asymptotes. Each runs from a corner (a,b) straight through the origin, so its slope is rise over run: The box is the picture; the equation is the proof. Solve for y on the right branch: Now push x toward infinity. Compared with the huge x^2 , the fixed -a^2 under the root becomes negligible, so the square root collapses onto x itself: That leaves exactly — the same slope the box gave us. And the leftover distance between branch and line never quite reaches zero: Take (so a=4 , b=3 ) with asymptote : At x=10 : branch , line y=7.50 — gap At x=100 : branch , line y=75.00 — gap At x=1000 : branch , line y=750.000 — gap The gap shrinks toward zero but is never exactly zero — the branch approaches its asymptote without ever crossing it. Which term is positive decides which way the hyperbola opens — and that flips the asymptote slope. Watch what happens to the very same numbers a=4 , b=3 when the roles of x and y swap.

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