x^2+6x+8y=7

Find the:
Direction:
Vertex:
Focus:
Axis of Symmetry:
Directrix: 

Oh, quadratic equations! Let's dive in and have some fun with your equation, x^2+6x+8y=7.

Direction: Well, since the coefficient of the x^2 term is positive, we know the parabola opens upward. So, the direction is up, up, and away!

Vertex: To find the vertex, we need to complete the square. We can rearrange the equation a bit to get it into the standard form (y = ax^2 + bx + c). Once we do that, we can use the formula h = -b/2a to find the x-coordinate of the vertex. The y-coordinate can be found by substituting the x-coordinate back into the equation. But be careful, you might find the quadratic equation hilarious!

Focus: The focus of the parabola can be found using the formula (h,k+p), where (h,k) is the vertex and p is a constant called the distance from the vertex to the focus. In this case, since the parabola opens upward, p will be positive.

Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex. In other words, it's a line of symmetry for our laughing parabola.

Directrix: The directrix is a horizontal line that's equidistant from each point on the parabola as the focus is. It's like a flat surface where our parabola bounces around in hilarious ways!

Now, go ahead and find those values yourself! And remember, math can be funny too!

To find the direction, vertex, focus, axis of symmetry, and directrix of the quadratic equation given by x^2+6x+8y=7, we first need to rewrite the equation in the standard form of a parabola.

1. Direction: Start by completing the square for the x terms.

x^2 + 6x + 8y = 7
x^2 + 6x + 9 = -8y + 7
(x^2 + 6x + 9) = -8(y - (7/8))

Since the coefficient of y is negative, the direction of the parabola is downward.

2. Vertex: The vertex of the parabola can be found by using the formula h = -b/2a, where a and b are the coefficients of x^2 and x, respectively.

In this case, a = 1 and b = 6, so the vertex can be calculated as follows:

h = -6/(2*1) = -6/2 = -3

Substituting h back into the equation, we can find the y-coordinate of the vertex.

x^2 + 6x + 9 = -8(y - (7/8))
(x + 3)^2 = -8(y - 7/8)

Comparing this with the standard form y = a(x - h)^2 + k, we can see that the vertex is (-3, 7/8).

3. Focus: The focus of the parabola can be found by using the formula (h, k + (1/4a)), where a is the coefficient of x^2.

In this case, a = 1, so the focus can be calculated as follows:

Focus = (-3, (7/8) + (1/(4*1)))
= (-3, 7/8 + 1/4)
= (-3, 7/8 + 2/8)
= (-3, 9/8)

Therefore, the focus of the parabola is (-3, 9/8).

4. Axis of Symmetry: The axis of symmetry of a parabola is a vertical line passing through the vertex. In this case, the axis of symmetry is x = -3.

5. Directrix: The directrix of the parabola can be found using the formula y = k - (1/4a), where a is the coefficient of x^2.

In this case, a = 1, so the directrix can be calculated as follows:

Directrix: y = (7/8) - (1/(4*1))
= (7/8) - (1/4)
= (7/8) - (2/8)
= (7 - 2)/8
= 5/8

Therefore, the directrix of the parabola is y = 5/8.

To find the direction, vertex, focus, axis of symmetry, and directrix of the given equation, let's first write the equation in the standard form of a parabola, which is (x - h)^2 = 4p(y - k):

x^2 + 6x + 8y = 7

To complete the square and rewrite the equation in the standard form, take half of the coefficient of x (which is 6/2 = 3) and square it (3^2 = 9). Add this value to both sides of the equation to maintain equality:

x^2 + 6x + 9 + 8y = 7 + 9

Rewrite the equation:

(x + 3)^2 + 8y = 16

Now, we can see that (x + 3)^2 = -8y + 16. Comparing this to the standard form (x - h)^2 = 4p(y - k), we find that h = -3 and k = -2.

So, the vertex (h, k) is (-3, -2).

To find the direction, we look at the coefficient of y. Since the coefficient of y is positive, the parabola opens upward. Hence, the direction is upward.

The axis of symmetry is a vertical line passing through the vertex. In this case, the axis of symmetry is x = -3.

To find the focus and directrix, we need to determine the value of p. Comparing the coefficient of y in the standard form and our equation, we see that 4p = 8, so p = 2.

The distance from the vertex to the focus is given by p, so the focus is located at (h, k + p). In this case, the focus is located at (-3, -2 + 2) = (-3, 0).

The distance from the vertex to the directrix is also given by p. Since the parabola opens upward, the directrix is a horizontal line located at a distance p below the vertex. Therefore, the directrix is the line y = -4.

In summary:
Direction: Upward
Vertex: (-3, -2)
Focus: (-3, 0)
Axis of Symmetry: x = -3
Directrix: y = -4

8y = -(x^2 + 6x - 7) = -(x^2 + 6x + 9) + 16

y = -1/8 (x + 3)^2 + 2 ... vertex form

-1/8 ... opens downward

vertex is (-3,2)
... midway between focus and directrix

focus is (-3,0)

axis of sym ... x = -3

directrix is ... y = 4