identify the type of curve and sketch the graph of y2-2x+6y+5=0
Assuming y2 means y^2, then we have a parabola, opening to the right:
2x = y^2 + 6y + 5
= (y+1)(y+5)
So it has zeroes at y=-1, y=-5, axis of symmetry at y = -3, vertex at (-2,-3).
To identify the type of curve and sketch the graph of the equation y^2 - 2x + 6y + 5 = 0, we need to rearrange the equation in a more recognizable form and analyze its properties.
First, let's rearrange the equation:
y^2 + 6y = 2x - 5
Now, complete the square for the y terms by adding (6/2)^2 = 9 to both sides of the equation:
y^2 + 6y + 9 = 2x - 5 + 9
(y + 3)^2 = 2x + 4
Now the equation is in the form (y + h)^2 = 4a(x - k), where (h, k) represents the vertex of the parabola.
Comparing this form with our equation, we can determine that h = 3, k = -2, and a = 1/2.
Now, let's analyze the properties of the curve:
1. Vertex: The vertex is (-2, 3), which represents the lowest point on the curve (since a > 0).
2. Axis of Symmetry: The axis of symmetry is given by x = -2.
3. Parabolic Nature: Since we have a squared term for y and a linear term for x, we can conclude that the equation represents a parabola.
4. Orientation: Since a > 0, the parabola opens upward.
To sketch the graph, we can use the properties we determined:
1. Plot the vertex at (-2, 3).
2. Draw a symmetric parabolic shape that opens upward.
3. Locate additional points on the graph by substituting different x values into the equation and calculating corresponding y values, or by using the symmetry of the parabola.
Based on the given information, you can sketch the graph of the equation y^2 - 2x + 6y + 5 = 0 as described.