At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To graph the focus and the directrix of the parabola given by the equation [tex]\(x = -\frac{1}{8}(y-3)^2 + 1\)[/tex], you need to start by identifying the vertex form of a parabola and using the properties of the parabola. Here's a step-by-step solution:
### Step 1: Identify the vertex and orientation
The standard form of a parabola that opens horizontally is given by:
[tex]\[ x = a(y - k)^2 + h \][/tex]
Comparing this with the given equation [tex]\( x = -\frac{1}{8}(y - 3)^2 + 1 \)[/tex], we can see that:
- The vertex [tex]\((h, k)\)[/tex] is [tex]\( (1, 3) \)[/tex]
- Since [tex]\(a = -\frac{1}{8}\)[/tex], the parabola opens to the left.
### Step 2: Determine the value of [tex]\(4p\)[/tex]
In the standard form [tex]\(x = a(y-k)^2 + h\)[/tex], [tex]\(a\)[/tex] is related to the parameter [tex]\(p\)[/tex] by:
[tex]\[ a = \frac{1}{4p} \][/tex]
From the given parabola:
[tex]\[ -\frac{1}{8} = \frac{1}{4p} \][/tex]
Solving for [tex]\(p\)[/tex]:
[tex]\[ 4p = -8 \][/tex]
[tex]\[ p = -2 \][/tex]
Here, [tex]\( p \)[/tex] tells us the distance from the vertex to the focus and from the vertex to the directrix along the axis of symmetry.
### Step 3: Find the focus
The focus is [tex]\(p\)[/tex] units away from the vertex in the direction the parabola opens. Since the parabola opens to the left and [tex]\(p = -2\)[/tex]:
- The focus is located 2 units to the left of the vertex: [tex]\((1, 3)\)[/tex].
[tex]\[ \text{Focus} = (1 - 2, 3) = (-1, 3) \][/tex]
### Step 4: Find the directrix
The directrix is a vertical line that is [tex]\(p\)[/tex] units away from the vertex in the opposite direction from the focus. Since [tex]\(p = -2\)[/tex], the directrix is 2 units to the right of the vertex at [tex]\( x = 3 \)[/tex]:
[tex]\[ \text{Directrix:} \, x = 1 - (-2) = 3 \][/tex]
### Step 5: Plot the parabola, vertex, focus, and directrix
1. Vertex: Plot the vertex at [tex]\((1, 3)\)[/tex].
2. Focus: Plot the focus at [tex]\((-1, 3)\)[/tex].
3. Directrix: Draw the vertical line [tex]\( x = 3 \)[/tex].
4. Parabola: Sketch the parabola opening to the left with the vertex at [tex]\((1, 3)\)[/tex], passing through points that satisfy the equation [tex]\( x = -\frac{1}{8}(y - 3)^2 + 1 \)[/tex].
### Final Result:
- Vertex: [tex]\((1, 3)\)[/tex]
- Focus: [tex]\((-1, 3)\)[/tex]
- Directrix: [tex]\( x = 3 \)[/tex]
Make sure to label these points and line clearly on your graph to indicate the focus and the directrix of the parabola.
### Step 1: Identify the vertex and orientation
The standard form of a parabola that opens horizontally is given by:
[tex]\[ x = a(y - k)^2 + h \][/tex]
Comparing this with the given equation [tex]\( x = -\frac{1}{8}(y - 3)^2 + 1 \)[/tex], we can see that:
- The vertex [tex]\((h, k)\)[/tex] is [tex]\( (1, 3) \)[/tex]
- Since [tex]\(a = -\frac{1}{8}\)[/tex], the parabola opens to the left.
### Step 2: Determine the value of [tex]\(4p\)[/tex]
In the standard form [tex]\(x = a(y-k)^2 + h\)[/tex], [tex]\(a\)[/tex] is related to the parameter [tex]\(p\)[/tex] by:
[tex]\[ a = \frac{1}{4p} \][/tex]
From the given parabola:
[tex]\[ -\frac{1}{8} = \frac{1}{4p} \][/tex]
Solving for [tex]\(p\)[/tex]:
[tex]\[ 4p = -8 \][/tex]
[tex]\[ p = -2 \][/tex]
Here, [tex]\( p \)[/tex] tells us the distance from the vertex to the focus and from the vertex to the directrix along the axis of symmetry.
### Step 3: Find the focus
The focus is [tex]\(p\)[/tex] units away from the vertex in the direction the parabola opens. Since the parabola opens to the left and [tex]\(p = -2\)[/tex]:
- The focus is located 2 units to the left of the vertex: [tex]\((1, 3)\)[/tex].
[tex]\[ \text{Focus} = (1 - 2, 3) = (-1, 3) \][/tex]
### Step 4: Find the directrix
The directrix is a vertical line that is [tex]\(p\)[/tex] units away from the vertex in the opposite direction from the focus. Since [tex]\(p = -2\)[/tex], the directrix is 2 units to the right of the vertex at [tex]\( x = 3 \)[/tex]:
[tex]\[ \text{Directrix:} \, x = 1 - (-2) = 3 \][/tex]
### Step 5: Plot the parabola, vertex, focus, and directrix
1. Vertex: Plot the vertex at [tex]\((1, 3)\)[/tex].
2. Focus: Plot the focus at [tex]\((-1, 3)\)[/tex].
3. Directrix: Draw the vertical line [tex]\( x = 3 \)[/tex].
4. Parabola: Sketch the parabola opening to the left with the vertex at [tex]\((1, 3)\)[/tex], passing through points that satisfy the equation [tex]\( x = -\frac{1}{8}(y - 3)^2 + 1 \)[/tex].
### Final Result:
- Vertex: [tex]\((1, 3)\)[/tex]
- Focus: [tex]\((-1, 3)\)[/tex]
- Directrix: [tex]\( x = 3 \)[/tex]
Make sure to label these points and line clearly on your graph to indicate the focus and the directrix of the parabola.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
