Math / Geometry

Question3. Determine a possible equation to represent each function. a)

Studdy Solution

STEP 1

1. The hyperbola is centered at the origin.
2. The asymptotes are the x-axis and y-axis.
3. The curves of the hyperbola are in the first and third quadrants.

STEP 2

1. Recall the standard form of a hyperbola equation centered at the origin.
2. Identify the orientation of the hyperbola.
3. Determine the equation based on the orientation.

STEP 3

Recall the standard form of a hyperbola equation centered at the origin. For a hyperbola with asymptotes along the axes, the equation is:
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
or
$\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$
depending on the orientation.

STEP 4

Identify the orientation of the hyperbola. Since the curves are in the first and third quadrants, the hyperbola opens along the x-axis. This means the equation takes the form:
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

STEP 5

Determine the equation based on the orientation. Since the asymptotes are the x-axis and y-axis, and the hyperbola is centered at the origin, a possible equation is:
$x^2 - y^2 = 1$
This assumes $a^2 = 1$ and $b^2 = 1$ for simplicity, as the exact values of $a$ and $b$ are not provided.
The possible equation for the hyperbola is:
$\boxed{x^2 - y^2 = 1}$

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