how do i graph a hyperbola from the following equation?
y^2-x^2-4y+4x-1+0
To graph a hyperbola, you can follow these steps:
Step 1: Write the equation in standard form
The equation of a hyperbola in standard form is given by:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1
For the given equation, we can rewrite it as:
(x^2 - 4x) - (y^2 + 4y) = 1
Step 2: Complete the Square
To complete the square, you need to add and subtract constant terms on both sides of the equation to make it as a perfect square trinomial.
(x^2 - 4x + ?) - (y^2 + 4y + ?) = 1 + ? + ?
The constant terms we need to add can be found by taking half of the coefficient of x (-4/2 = -2) and y (4/2 = 2), and squaring them to get 4 and 1 respectively.
(x^2 - 4x + 4) - (y^2 + 4y + 4) = 1 + 4 - 4
Simplifying, we have:
(x - 2)^2 - (y + 2)^2 = 1
Step 3: Identify the center and the vertices
The center of the hyperbola is given by the values of (h, k) in the standard form. In this case, the center is (2, -2).
The distance from the center to the vertices along the x-axis is given by 'a', which is the square root of the value after (x - h)^2. In this case, a = √1 = 1.
The distance from the center to the vertices along the y-axis is given by 'b', which is the square root of the value after (y - k)^2. In this case, b = √1 = 1.
So, the vertices are located at (2 ± a, -2), which gives us (3, -2) and (1, -2).
Step 4: Plot the center, vertices, and asymptotes
Plot the center (2, -2) on the graph. Then, draw the asymptotes, which intersect the center and pass through the vertices.
The equation of the asymptotes for a hyperbola can be found using the formula: "y - k = ± (b/a)(x - h)". In this case, the asymptotes are y + 2 = ±(1/1)(x - 2), which simplifies to y = x - 4 and y = x.
Step 5: Draw the hyperbola
The hyperbola can be sketched by drawing the branches based on the center, vertices, and asymptotes. The branches extend outward from the center along the x and y axes.
So, plot points along the branches using symmetrical values and sketch the curve. In this case, you can plot points like (0, -2), (1, -3), (3, -1), (4, 0), etc.
By connecting all the points, you will obtain the graph of the hyperbola.
Remember to label the axes and add any other important details to the graph.
I hope this helps you in graphing the given hyperbola!