How do you sketch an equation like

x+4=-2(y-3)^2

To understand how to sketch the equation x + 4 = -2(y - 3)^2, we will go through a step-by-step process.

1. Rewrite the equation in standard form:
Start by expanding the equation. To do this, use the distributive property to multiply -2 by (y - 3)^2, and then arrange the terms so the equation is in the form y = ...:
x + 4 = -2(y - 3)^2
x + 4 = -2(y^2 - 6y + 9)
x + 4 = -2y^2 + 12y - 18
Rearrange the equation to have zero on one side:
2y^2 - 12y - x + 22 = 0

2. Identify the vertex and axis of symmetry:
To sketch the equation, we will use the vertex form of a parabola: y = a(x - h)^2 + k. The vertex form provides information about the vertex and axis of symmetry.
In our equation, 2y^2 - 12y - x + 22 = 0, we can rewrite it as:
2(y^2 - 6y) = x - 22
2(y^2 - 6y + 9) = x - 22 + 18 (Completing the square by adding 9 to both sides)
2(y - 3)^2 = x - 4
(y - 3)^2 = (1/2)(x - 4)
Comparing this to y = a(x - h)^2 + k, we can see that h = 4 and k = 3. So, the vertex is (4, 3), and the axis of symmetry is the vertical line x = 4.

3. Determine the orientation of the parabola:
The coefficient of x in the vertex form gives information about the orientation of the parabola. If the coefficient is positive, the parabola opens upward, and if it's negative, the parabola opens downward.
In our equation, the coefficient of x is positive (1/2), so the parabola opens upward.

4. Plot the vertex and the axis of symmetry:
On a graph, plot the vertex (4, 3) and draw a vertical line passing through it, representing the axis of symmetry x = 4.

5. Determine additional points:
To plot additional points, choose some x-values and substitute them into the equation to find the corresponding y-values. Since we have an equation in terms of y, solving for y and finding x will be a bit more complex. Instead, we can consider y as our input variable and find corresponding x-values.

For example, let's choose values for y and solve for x:
- Let y = 4:
(y - 3)^2 = (1/2)(x - 4)
(4 - 3)^2 = (1/2)(x - 4)
1 = (1/2)(x - 4)
2 = x - 4
x = 6
So, we have a point (6, 4).

- Let y = 2:
(2 - 3)^2 = (1/2)(x - 4)
(-1)^2 = (1/2)(x - 4)
1 = (1/2)(x - 4)
2 = x - 4
x = 6
So, we have another point (6, 2).

Choose a few more values of y and calculate the corresponding x-values to get additional points.

6. Plot the points and sketch the parabola:
Using the vertex (4, 3), the axis of symmetry x = 4, and the additional points found in the previous step, plot the points and sketch the parabola that passes through them. Ensure that the parabola is symmetric about the axis of symmetry x = 4 and opens upward.

By following these steps, you can sketch the equation x + 4 = -2(y - 3)^2 on a graph.