write the vertex form equation of each parabola.

1) Vertex:(-5,8), Focus:(-21/4, 8)
2) Vertex:(-6,-9), Directrix: x= 47/8
3)Vertex(8,-1) y- intercept: -17
4) Open left or right, Vertex: (7, 6), passes through:(-11,9)
5)Focus(-63/8, -7), Directrix: x= -65/8
6 Opens up or down, and passes through (−6, −7), (−11, −2), and (−8, 1)
7) Vertex at origin, opens left, 1/8units between the vertex and focus.
8) Vertex: (10, 0), axis of symmetry: y = 0,
length of latus rectum = 1, a < 0

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To get you started, the parabola
y^2 = 4px has
vertex at (0,0)
focus at (0,p)
directrix is the line x = -p
latus rectum has length 4p

So, for
5)Focus(-63/8, -7), Directrix: x= -65/8
The distance between focus and directrix is 2p, so 2p = 1/4, p = 1/8
Since the directrix is a vertical line, the axis is horizontal
Since the focus is at y = -7, that is the axis
The vertex is at (-64/8,-7) = (-8,-7)
So the equation is
(y+7)^2 = 1/2 (x+8)
See the graph and properties at

https://www.wolframalpha.com/input/?i=parabola+%28y%2B7%29%5E2+%3D+1%2F2+%28x%2B8%29

it suppose to be a homework help page no?

To write the vertex form equation of a parabola, you can use the formula:

y = a(x - h)^2 + k, where (h, k) represents the vertex coordinates.

1) Given: Vertex: (-5,8), Focus: (-21/4, 8)
To find 'a' (the coefficient of x^2), we need to calculate the distance between the vertex and the focus. The formula for the distance is:
Distance = 1 / (4a)

Distance = sqrt((h - p)^2 + (k - q)^2)
Solving for 'a', we get:
1 / (4a) = sqrt((-5 + 21/4)^2)
1 / (4a) = sqrt((-20/4 + 21/4)^2)
1 / (4a) = sqrt((1/4)^2)
1 / (4a) = 1/4
1 / (a) = 1
a = 1

Now we can substitute the values into the vertex form equation:
y = (x + 5)^2 + 8

2) Given: Vertex: (-6,-9), Directrix: x = 47/8
To write the equation, we need to find the value of 'a'.

The distance between the vertex and the directrix is equal to the absolute value of 'a'.
Distance = |a|

Vertex = (-6, -9), Directrix = x = 47/8

Distance = |a| = |-9 - (47/8)|
Distance = |-9 - (47/8)| = |(-72/8) - (47/8)|
Distance = |(-72 - 47)/8| = |-119/8|
a = -119/8

Now substitute the values into the vertex form equation:
y = (-119/8)(x + 6)^2 - 9

3) Given: Vertex: (8,-1), y-intercept: -17
To find 'a', we can substitute the given points into the vertex form equation.

y = a(x - 8)^2 - 1
Substitute (-17, 0) into the equation:
0 = a(8 - 8)^2 - 1
0 = a(0)^2 - 1
0 = -1

Therefore, there is no 'a' value, making the equation: y = -1

4) Given: Open left or right, Vertex: (7, 6), passes through: (-11, 9)
To determine the direction and find 'a', we can substitute the vertex coordinates into the vertex form equation.

Since the axis of symmetry is vertical, the parabola is opening left for x < 7:
y = a(x - 7)^2 + 6

Now use the given point (-11, 9) to find 'a':
9 = a(-11 - 7)^2 + 6
9 = a(-18)^2 + 6
9 = 324a + 6
3 = 324a
a = 3/324
a = 1/108

Adding the value of 'a' to the equation, we get:
y = (1/108)(x - 7)^2 + 6

5) Given: Focus: (-63/8, -7), Directrix: x = -65/8
Using the same process as before, we need to find 'a' by calculating the distance between the focus and the vertex.

Focus = (-63/8, -7), Directrix = x = -65/8

Distance = |a| = |-63/8 - (-65/8)|
Distance = |-63/8 + 65/8| = |(-63 + 65)/8|
Distance = |2/8| = |1/4|
a = 1/4

Now, substitute the values into the vertex form equation:
y = (1/4)(x + 63/8)^2 - 7

6) Given: Opens up or down, and passes through (-6, -7), (-11, -2), and (-8, 1)
Since the parabola passes through three points, we can substitute each point into the vertex form equation to find 'a'.

Substitute the first point (-6, -7) into the equation:
-7 = a(-6 - h)^2 + k
-7 = a(-6 + 6)^2 + k
-7 = 0 + k
k = -7

Substitute the second point (-11, -2) into the equation:
-2 = a(-11 - h)^2 - 7
-2 = a(-11 + 6)^2 - 7
-2 = a(-5)^2 - 7
-2 = 25a - 7
25a = 5
a = 1/5

Substitute the third point (-8, 1) into the equation:
1 = a(-8 - h)^2 - 7
1 = a(-8 + 6)^2 - 7
1 = 4a - 7
4a = 8
a = 2

Since the value of 'a' is inconsistent among the three points, it seems incorrect or contradictory.

7) Given: Vertex at origin, opens left, 1/8 units between the vertex and the focus.
Since the parabola opens left, the equation is of the form: x = a(y - k)^2.

Since there is a 1/8 unit between the vertex and the focus, the value of 'a' can be calculated by:
a = 1 / (4h)

Given that the vertex is at the origin (0, 0), we have:
a = 1 / (4 * 0)
a = 1 / 0

The value of 'a' is undefined, implying that there is no solution for this question.

8) Given: Vertex: (10, 0), Axis of symmetry: y = 0, Length of latus rectum = 1, a < 0
The vertex form equation for this parabola is y = a(x - h)^2 + k.
Since the axis of symmetry is y = 0, we get y = ax^2 + k.

Given that the length of latus rectum is 1, we know that:
4a = 1
a = 1/4

Substituting the value of 'a' into the equation, we get:
y = (1/4)x^2 + k

Since the vertex is at (10, 0), we can substitute these coordinates into the equation to find 'k':
0 = (1/4)(10)^2 + k
0 = 25/2 + k
k = -25/2

The equation for this parabola is:
y = (1/4)x^2 - 25/2

Try and you will get help if stuck.

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