Give the standard formula of the parabola with a vertex (-1.5,4)and a focus (-4,4)

To find the standard formula of a parabola given the vertex and the focus, we need to determine the orientation and use the formula for a parabola in the form:

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

or

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

where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus. 

### Step-by-Step Solution

1. **Determine the orientation of the parabola:**

   The vertex is \((-1.5, 4)\) and the focus is \((-4, 4)\). Since both the vertex and the focus have the same \(y\)-coordinate, the parabola opens horizontally. Because the focus is to the left of the vertex, the parabola opens to the left.

2. **Use the standard form of a horizontally opening parabola:**

   The equation of a parabola that opens horizontally is:

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

3. **Identify the vertex \((h, k)\):**

   Given vertex: \((h, k) = (-1.5, 4)\).

4. **Calculate the distance \(p\) between the vertex and the focus:**

   Since the parabola opens horizontally, \(p\) is the horizontal distance between the vertex and the focus. 

   The focus is at \((-4, 4)\), so:

   \[
p = -4 - (-1.5) = -4 + 1.5 = -2.5
   \]

   Since \(p\) is negative (because the parabola opens to the left), we have \(p = -2.5\).

5. **Write the equation of the parabola:**

   Substitute \((h, k) = (-1.5, 4)\) and \(p = -2.5\) into the standard form:

   \[
(y - 4)^2 = 4(-2.5)(x + 1.5)
\]

   Simplify the equation:

   \[
(y - 4)^2 = -10(x + 1.5)
\]

### Final Answer

The standard equation of the parabola with vertex \((-1.5, 4)\) and focus \((-4, 4)\) is:

\[
(y - 4)^2 = -10(x + 1.5).
\]

Would you like more details on this process, or do you have any questions?

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Here are 5 related questions to consider:

1. How do you find the directrix of a parabola given its vertex and focus?
2. What is the effect of the distance \(p\) on the shape of the parabola?
3. How would you find the equation of a parabola that opens vertically instead of horizontally?
4. How can you determine if a parabola opens upwards, downwards, to the left, or to the right?
5. What are the properties of a parabola in the standard form compared to the vertex form?

**Tip:** Remember, the sign of \(p\) in a parabola's equation determines the direction in which it opens!

Discover how to find the standard formula of a parabola given its vertex and focus. This problem involves coordinate geometry and understanding the shape of parabolas. Suitable for high school students.

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