# Given the equation x^2 + 6x + 4y - 7 = 0 determine: a)  The Vertex V b)  If the parabola opens up, down, left, right c)  The Focus F

samhouston | Middle School Teacher | (Level 1) Associate Educator

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First, solve the equation for y.

x^2 + 6x + 4y - 7 + (-4y) = 0 + (-4y)
x^2 + 6x - 7 = -4y
Divide both sides by -4.
-0.25x^2 + -1.5x + 1.75 = y

Second, identify the values of a, b, and c.
y = ax^2 + bx + c
a = -0.25
b = -1.5
c = 1.75

Next, calculate the vertex's x-value using the formula:
x = -b/2a
x = -(-1.5) / 2 * (-0.25)
x = 1.5 / (-0.5)
x = -3

Now substiute -3 in for x and solve for y.  This will give you the vertex's y-value.
(-0.25) * -3^2 + (-1.5) * -3 + 1.75 = y
(-0.25) * 9 + 4.5 + 1.75 = y
(-2.25) + 4.5 + 1.75 = y
y = 4

The vertex is (-3, 4).

Since the value of a is negative, the parabola opens down.

The focus of a parabola is a fixed point on the interior of the parabola.  Since the parabola opens down, the x-value of the focus is the same as the x-value of the vertex.  To find the y-value of the focus, use this formula:
y = c - (b^2 - 1)/4a
y = 1.75 - (-1.5^2 - 1) / 4 * -0.25
y = 1.75 - (2.25 - 1) / 4 * -0.25
y = 1.75 - 1.25 / -1
y = 1.75 + 1.25
y = 3

The focus is (-3, 3).

Here is a graph of the parabola:

Summary:

a.  Vertex = (-3, 4)

b.  Parabola opens down

c.  Focus = (-3, 3)