Gráfica de la Hipérbola dada su ecuación canónica - ordinaria | Ejemplo 2

Summary

The instructor demonstrates how to graph a hyperbola in canonical form when its center is at the origin. The main example is

[ \frac{x^2}{16}-\frac{y^2}{9}=1. ]

The key idea is to identify the positive term first: its variable indicates the direction of the hyperbola’s principal (transverse) axis. The instructor then calculates the hyperbola’s dimensions and plots its center, vertices, foci, and latus rectum to guide the sketch.

Method

1. Identify the axis direction

The positive term contains (x), so the principal axis is parallel to the (x)-axis. Therefore, (a) and (c) are measured horizontally, while (b) is measured vertically.

2. Read the center and parameters

There are no shifts in (x) or (y), so the center is ((0,0)). From the equation,

[ a^2=16,\qquad b^2=9, ]

so (a=4) and (b=3).

3. Find the focal distance

For a hyperbola,

[ c^2=a^2+b^2. ]

Thus,

[ c=\sqrt{16+9}=5. ]

These values are positive because they represent distances.

4. Find the latus rectum length

The latus rectum length is

[ L=\frac{2b^2}{a}. ]

For this example,

[ L=\frac{2(9)}{4}=4.5. ]

The latus rectum is perpendicular to the principal axis, passes through a focus, and has length (4.5).

5. Plot the points and sketch the hyperbola

• Plot the center at ((0,0)).
• Plot the vertices four units to the left and right: ((-4,0)) and ((4,0)).
• Plot the foci five units to the left and right: ((-5,0)) and ((5,0)).
• Use (b=3) to form a guide rectangle with corners ((-4,3)), ((-4,-3)), ((4,3)), and ((4,-3)). The rectangle helps guide the asymptotes.
• At each focus, mark the latus rectum endpoints (2.25) units above and below it, since half of (4.5) is (2.25).
• Draw the two branches through the vertices and latus rectum endpoints. The asymptote equations are not derived in this lesson.

Practice Example

The instructor also works through a hyperbola with the same orientation, where (a^2=9) and (b^2=4). This gives

[ a=3,\qquad b=2,\qquad c=\sqrt{13}\approx 3.6, ]

and a latus rectum length of

[ \frac{8}{3}\approx 2.6. ]

The same plotting procedure applies.

Speaker and Source

Matemáticas profe Alex is the instructor and sole speaker featured.