Math / Geometry

QuestionTicket: Identify the following the graph: vertex: $(4,-1)$ Focus: $(4,1)$ P -value: 5 Directrix: $y=-9$

Studdy Solution

STEP 1

1. The vertex, focus, and directrix are given for a parabola.
2. The vertex form of a parabola is used to identify the equation.
3. The parabola opens vertically since the x-coordinates of the vertex and focus are the same.

STEP 2

1. Understand the relationship between the vertex, focus, and directrix.
2. Determine the orientation and direction of the parabola.
3. Use the vertex form of the parabola to write the equation.
4. Verify the equation with given values.

STEP 3

Understand the relationship between the vertex, focus, and directrix:
The vertex is the midpoint between the focus and the directrix. The parabola opens towards the focus. Given: - Vertex: $(4, -1)$ - Focus: $(4, 1)$ - Directrix: $y = -9$

STEP 4

Determine the orientation and direction of the parabola:
Since the x-coordinates of the vertex and focus are the same, the parabola opens vertically. The focus is above the vertex, so the parabola opens upwards.

STEP 5

Use the vertex form of the parabola to write the equation:
The vertex form for a vertical parabola is:
$(x - h)^2 = 4p(y - k)$
where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus.
Given: - Vertex: $(h, k) = (4, -1)$ - $p = 1 - (-1) = 2$
Substitute these values into the vertex form:
$(x - 4)^2 = 4 \times 2 (y + 1)$ $(x - 4)^2 = 8(y + 1)$

STEP 6

Verify the equation with given values:
Check that the focus and directrix satisfy the equation:
- The focus $(4, 1)$ should satisfy $(x - 4)^2 = 8(y + 1)$ :
$(4 - 4)^2 = 8(1 + 1)$ $0 = 16$
- The directrix $y = -9$ is consistent with the vertex form since $y = -9$ is below the vertex.
The equation of the parabola is:
$\boxed{(x - 4)^2 = 8(y + 1)}$

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QuestionTicket: Identify the following the graph: vertex: $(4,-1)$ Focus: $(4,1)$ P -value: 5 Directrix: $y=-9$

Studdy Solution

STEP 1

1. The vertex, focus, and directrix are given for a parabola.
2. The vertex form of a parabola is used to identify the equation.
3. The parabola opens vertically since the x-coordinates of the vertex and focus are the same.

STEP 2

1. Understand the relationship between the vertex, focus, and directrix.
2. Determine the orientation and direction of the parabola.
3. Use the vertex form of the parabola to write the equation.
4. Verify the equation with given values.

STEP 3

Understand the relationship between the vertex, focus, and directrix:
The vertex is the midpoint between the focus and the directrix. The parabola opens towards the focus. Given: - Vertex: $(4, -1)$ - Focus: $(4, 1)$ - Directrix: $y = -9$

STEP 4

Determine the orientation and direction of the parabola:
Since the x-coordinates of the vertex and focus are the same, the parabola opens vertically. The focus is above the vertex, so the parabola opens upwards.

STEP 5

STEP 6

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QuestionTicket: Identify the following the graph: vertex: (4,−1)(4,-1)(4,−1) Focus: (4,1)(4,1)(4,1) P -value: 5 Directrix: y=−9y=-9y=−9

Studdy Solution

STEP 1

1. The vertex, focus, and directrix are given for a parabola.2. The vertex form of a parabola is used to identify the equation.3. The parabola opens vertically since the x-coordinates of the vertex and focus are the same.

STEP 2

1. Understand the relationship between the vertex, focus, and directrix.2. Determine the orientation and direction of the parabola.3. Use the vertex form of the parabola to write the equation.4. Verify the equation with given values.

STEP 3

Understand the relationship between the vertex, focus, and directrix:The vertex is the midpoint between the focus and the directrix. The parabola opens towards the focus. Given: - Vertex: (4,−1) (4, -1) (4,−1) - Focus: (4,1) (4, 1) (4,1) - Directrix: y=−9 y = -9 y=−9

STEP 4

Determine the orientation and direction of the parabola:Since the x-coordinates of the vertex and focus are the same, the parabola opens vertically. The focus is above the vertex, so the parabola opens upwards.

STEP 5

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionTicket: Identify the following the graph: vertex: $(4,-1)$ Focus: $(4,1)$ P -value: 5 Directrix: $y=-9$

1. The vertex, focus, and directrix are given for a parabola.
2. The vertex form of a parabola is used to identify the equation.
3. The parabola opens vertically since the x-coordinates of the vertex and focus are the same.

1. Understand the relationship between the vertex, focus, and directrix.
2. Determine the orientation and direction of the parabola.
3. Use the vertex form of the parabola to write the equation.
4. Verify the equation with given values.

Understand the relationship between the vertex, focus, and directrix:
The vertex is the midpoint between the focus and the directrix. The parabola opens towards the focus. Given: - Vertex: $(4, -1)$ - Focus: $(4, 1)$ - Directrix: $y = -9$

Determine the orientation and direction of the parabola:
Since the x-coordinates of the vertex and focus are the same, the parabola opens vertically. The focus is above the vertex, so the parabola opens upwards.