Ellipses and Hyperbolas

Hyperbola Asymptotes

Find the asymptote line equations from a centered horizontal hyperbola in standard form.

hyperbola-asymptotes For x^2/a^2 - y^2/b^2 = 1, the asymptote slopes are b/a and -b/a.

Example

Use b over a to write and plot the asymptotes.

highlighted = computed this step

Step 1 — Read a and b

Read a and b for the hyperbola.

a=3b=4a= \hl{3} \quad b= \hl{4}

Step 2 — Find asymptote slope

Divide b by a for the asymptote slope.

m=4/3=43m= 4 / 3 = \hlmath{\frac{4}{3}}

Step 3 — Asymptote equations

Write both asymptote equations.

y=43xy=−43xy= \hlmath{\frac{4}{3}} x \quad y=- \hlmath{\frac{4}{3}} x

Step 4 — Plot asymptotes

Plot the hyperbola and asymptotes.

Coordinate plot\text{Coordinate plot}
Hyperbola asymptotes plotCentered horizontal hyperbola with its two asymptote lines.updown

Step 5 — Slopes

State the opposite asymptote slopes.

m=43,−43m= \hlmath{\frac{4}{3}} , - \hlmath{\frac{4}{3}}