Analyze: foci, vertices, asymptotes, center, transverse (major) axis. Then graph.
2x^2-y^2+4x+4y-4=0
2 x^2 + 4 x = y^2 - 4 y +4
2 x^2 + 4 x = (y-2)^2
x^2 + 2 x - (1/2) (y-2)^2 = 0
x^2 + 2 x + 1 - (1/2) (y-2)^2 = 1
(x+1)^2 - (y-2)^2/(sqrt 2)^2 = 1
Hyperbola, you look up the stuff for it.
To analyze the given equation and find the foci, vertices, asymptotes, center, and transverse (major) axis, we need to rewrite it in a standard form for a hyperbola.
Step 1: Group the terms:
2x^2 + 4x - y^2 + 4y - 4 = 0
Step 2: Complete the square for x terms (group x's together):
2(x^2 + 2x) + (4y - y^2 - 4) = 4
Step 3: Complete the square for y terms (group y's together):
2(x^2 + 2x) + (-1)(y^2 - 4y) = 4
Step 4: Factor out coefficients of x and y terms:
2(x^2 + 2x) - 1(y^2 - 4y) = 4
Step 5: Now we need to add and subtract the necessary values to complete the square. For x terms, we need to add (2/2)^2 = 1, and for y terms, we need to add (-4/2)^2 = 4.
2(x^2 + 2x + 1) - 1(y^2 - 4y + 4) = 4 + 2 - 4
Step 6: Simplify the equation:
2(x + 1)^2 - (y - 2)^2 = 2
Step 7: Divide both sides by the constant on the right side to normalize the equation:
(x + 1)^2/1 - (y - 2)^2/2 = 1
Step 8: Compare the equation with the standard form for a hyperbola:
(x - h)^2/a^2 – (y - k)^2/b^2 = 1
From the comparison, we can determine the following parameters:
- Center: (h, k) = (-1, 2)
- a^2 = 1, so a = 1 (semi-major axis)
- b^2 = 2, so b = sqrt(2) (semi-minor axis)
Step 9: Find the foci and vertices:
The distance between the foci (c) and the center is given by c^2 = a^2 + b^2.
c^2 = 1^2 + sqrt(2)^2
c^2 = 1 + 2
c^2 = 3
c = sqrt(3)
- Foci: The foci are located at (-1 ± sqrt(3), 2), so the coordinates are approximately (-1.73, 2) and (0.73, 2).
- Vertices: The coordinates of the vertices are (-1 ± a, 2), so the coordinates are (-2, 2) and (0, 2).
Step 10: Find the asymptotes:
The equations of the asymptotes for a hyperbola in standard form are given by y = ± (b/a)(x - h) + k.
Substituting the values, we have:
y = ± (sqrt(2)/1)(x + 1) + 2
Simplifying further:
y = ± sqrt(2)(x + 1) + 2
Now we are ready to graph the hyperbola:
- Plot the center at (-1, 2).
- Plot the foci at (-1.73, 2) and (0.73, 2).
- Plot the vertices at (-2, 2) and (0, 2).
- Draw the asymptotes passing through the center and extending to infinity.
Note: Since the equation represents a hyperbola, the branches will be opening left and right.