Graph (x^2)/4-(y^2)/4=1

Problem

(x^2)/4−(y^2)/4=1

Solution

1. Identify the type of conic section. Since the equation is in the form (x^2)/(a^2)−(y^2)/(b^2)=1 it represents a horizontal hyperbola centered at the origin (0,0)

2. Determine the values of a and b Comparing the given equation to the standard form, we find a^2=4 and b^2=4 which means a=2 and b=2

3. Locate the vertices. For a horizontal hyperbola, the vertices are at (±a,0) Substituting a=2 the vertices are (2,0) and (−2,0)

4. Find the asymptotes. The equations for the asymptotes of a hyperbola centered at the origin are y=±b/a*x Substituting a=2 and b=2 gives y=±2/2*x which simplifies to y=x and y=−x

5. Calculate the foci. The distance from the center to the foci is c where c^2=a^2+b^2 Here, c^2=4+4=8 so c=√(,8)=2√(,2) The foci are at (±2√(,2),0)

6. Sketch the graph. Plot the vertices and the asymptotes. Draw two curves opening to the left and right, starting from the vertices and approaching the asymptotes.

Final Answer

(x^2)/4−(y^2)/4=1* is a hyperbola with vertices *(±2,0)* and asymptotes *y=±x

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