vertex from when the vertex is (6,6) and a point is (61/10,3/2)

To find the equation of a parabola given its vertex and a point on the curve, we use the **vertex form** of the equation of a parabola:

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

where:
- \((h, k)\) is the vertex of the parabola,
- \(a\) is a constant that determines the shape (how wide or narrow it is),
- \((x, y)\) is any point on the parabola.

Given:
- Vertex \((h, k) = (6, 6)\),
- Point \((x, y) = \left(\frac{61}{10}, \frac{3}{2}\right)\).

### Step 1: Substitute the vertex into the vertex form
\[
y = a(x - 6)^2 + 6
\]

### Step 2: Substitute the point \(\left(\frac{61}{10}, \frac{3}{2}\right)\) into the equation to solve for \(a\)

\[
\frac{3}{2} = a\left(\frac{61}{10} - 6\right)^2 + 6
\]

First, simplify the expression inside the parentheses:

\[
\frac{61}{10} - 6 = \frac{61}{10} - \frac{60}{10} = \frac{1}{10}
\]

Now, plug this into the equation:

\[
\frac{3}{2} = a\left(\frac{1}{10}\right)^2 + 6
\]
\[
\frac{3}{2} = a\left(\frac{1}{100}\right) + 6
\]

### Step 3: Solve for \(a\)

Subtract 6 from both sides:

\[
\frac{3}{2} - 6 = a \cdot \frac{1}{100}
\]
\[
\frac{3}{2} - \frac{12}{2} = a \cdot \frac{1}{100}
\]
\[
-\frac{9}{2} = a \cdot \frac{1}{100}
\]

Multiply both sides by 100:

\[
-450 = a
\]

### Step 4: Write the equation of the parabola

Now that we have \(a = -450\), the equation of the parabola is:

\[
y = -450(x - 6)^2 + 6
\]

This is the equation of the parabola with vertex \((6, 6)\) passing through the point \(\left(\frac{61}{10}, \frac{3}{2}\right)\).

---

Would you like any further details or clarification on the steps?

Here are 5 related questions to consider:
1. How can you determine the direction a parabola opens from its equation?
2. What is the significance of the value of \(a\) in the vertex form?
3. How would you find the focus and directrix of this parabola?
4. What is the relationship between the vertex form and standard form of a parabola?
5. How does changing the vertex affect the shape of the parabola?

**Tip:** When solving for \(a\) in a parabola's vertex form, the closer the point is to the vertex, the more accurate your calculation will be.

Find the equation of the parabola with vertex (6, 6) and passing through the point (61/10, 3/2).

Learn how to find the equation of a parabola with vertex (6, 6) and a point (61/10, 3/2). This step-by-step solution uses the vertex form of a quadratic equation and demonstrates how to determine the value of 'a'. Suitable for high school students in Grades 9-12.

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