Identify vertex, focus, axis of symmetry and directrix of the following  8(y+3)=(x-4)

To identify the vertex, focus, axis of symmetry, and directrix of the parabola given by the equation:

\[
8(y + 3) = (x - 4)
\]

we first need to rewrite this equation in a standard form for a parabola. The standard form for a parabola that opens horizontally is:

\[
(y - k)^2 = 4p(x - h)
\]

### Step-by-Step Solution

1. **Rewrite the equation in standard form:**

   Given equation:  
   \[
   8(y + 3) = (x - 4)
   \]

   Divide both sides by 8 to isolate \((y + 3)\):

   \[
   y + 3 = \frac{1}{8}(x - 4)
   \]

   Now, rearrange to match the standard form \((y - k)^2 = 4p(x - h)\):

   \[
   (y - (-3))^2 = 4p(x - 4)
   \]

   Notice that here \((y + 3)^2\) is the squared term; so, we rewrite the equation considering the squared term as a product to make it fit the parabola's equation format.

2. **Identifying components:**

   Comparing with the standard form \((y - k)^2 = 4p(x - h)\), we get:
   - \(h = 4\)
   - \(k = -3\)
   - \(4p = \frac{1}{8}\)

   Solving for \(p\):
   \[
   4p = \frac{1}{8} \implies p = \frac{1}{32}
   \]

3. **Identify the vertex:**

   The vertex \((h, k)\) of the parabola is:
   \[
   (h, k) = (4, -3)
   \]

4. **Identify the focus:**

   For a horizontally oriented parabola that opens to the right (if \(p > 0\)) or to the left (if \(p < 0\)), the focus is \((h + p, k)\).

   Here, \(p = \frac{1}{32}\), which is positive, indicating the parabola opens to the right.

   Focus:
   \[
   (h + p, k) = \left(4 + \frac{1}{32}, -3\right) = \left(\frac{129}{32}, -3\right)
   \]

5. **Identify the axis of symmetry:**

   The axis of symmetry for a horizontal parabola is the horizontal line through the vertex:
   \[
   y = k
   \]
   So, the axis of symmetry is:
   \[
   y = -3
   \]

6. **Identify the directrix:**

   The directrix for a horizontally oriented parabola is a vertical line \(x = h - p\).

   Directrix:
   \[
   x = 4 - \frac{1}{

Learn how to identify the vertex, focus, axis of symmetry, and directrix of a parabola with a detailed step-by-step solution. This problem covers key concepts in parabolas, including the vertex form, focus, and directrix. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation