Graph y^2-(x^2)/9=1

Problem

y^2−(x^2)/9=1

Solution

1. Identify the type of conic section by looking at the equation. Since the terms y^2 and x^2 have opposite signs, the equation represents a hyperbola.

2. Determine the orientation of the hyperbola. Because the y^2 term is positive, the hyperbola opens vertically (upward and downward).

3. Find the vertices by setting x=0 This gives y^2=1 so the vertices are at (0,1) and (0,−1)

4. Identify the values of a and b from the standard form (y^2)/(a^2)−(x^2)/(b^2)=1 Here, a^2=1 (so a=1 and b^2=9 (so b=3.

5. Determine the equations of the asymptotes using the formula y=±a/b*x Substituting the values gives y=±1/3*x

6. Sketch the graph by plotting the vertices, drawing the asymptotes, and drawing the two branches of the hyperbola approaching the asymptotes from the vertices.

Final Answer

y^2−(x^2)/9=1* is a vertical hyperbola with vertices *(0,±1)* and asymptotes *y=±1/3*x

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