vertex form when vertex is (-1/4, 3/2) and a point is (-2,0)

To find the **vertex form of a quadratic equation**, we use the following formula:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola, and \(a\) is a constant that determines how wide or narrow the parabola is. 

Given:
- Vertex: \( \left(-\frac{1}{4}, \frac{3}{2} \right) \)
- A point on the parabola: \( (-2, 0) \)

### Step 1: Plug in the vertex into the vertex form.
\[
y = a\left(x + \frac{1}{4}\right)^2 + \frac{3}{2}
\]
Now, we need to find the value of \(a\) using the point \( (-2, 0) \).

### Step 2: Substitute the point \( (-2, 0) \) into the equation.
\[
0 = a\left(-2 + \frac{1}{4}\right)^2 + \frac{3}{2}
\]

Simplify inside the parentheses:
\[
-2 + \frac{1}{4} = -\frac{8}{4} + \frac{1}{4} = -\frac{7}{4}
\]
So:
\[
0 = a\left(-\frac{7}{4}\right)^2 + \frac{3}{2}
\]

### Step 3: Simplify the equation.
\[
\left(-\frac{7}{4}\right)^2 = \frac{49}{16}
\]
\[
0 = a \cdot \frac{49}{16} + \frac{3}{2}
\]

### Step 4: Solve for \(a\).
First, subtract \(\frac{3}{2}\) from both sides:
\[
-\frac{3}{2} = a \cdot \frac{49}{16}
\]

Now, multiply both sides by \( \frac{16}{49} \) to isolate \(a\):
\[
a = -\frac{3}{2} \cdot \frac{16}{49} = -\frac{48}{98} = -\frac{24}{49}
\]

### Step 5: Write the final equation.
Now that we know \(a = -\frac{24}{49}\), the vertex form of the equation is:

\[
y = -\frac{24}{49}\left(x + \frac{1}{4}\right)^2 + \frac{3}{2}
\]

---

Do you need further details or clarification?  

Here are five related questions:  
1. How do you convert vertex form to standard form?  
2. How do you determine the direction of a parabola using \(a\)?  
3. What does the vertex of a quadratic function represent?  
4. How can you find the x-intercepts from vertex form?  
5. How do you complete the square to transform a standard form equation into vertex form?

**Tip:** The sign of \(a\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

Explore how to find the vertex form of a parabola when given the vertex (-1/4, 3/2) and a point (-2, 0). This problem demonstrates how to apply the vertex form equation of a quadratic function and solve for the constant 'a'. Suitable for students in Grades 10-12.

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