https://file.aiquickdraw.com/mathbot/file/1727457875325axqhlle1.jpg

The question asks for the equation of a parabola in vertex form, given that its vertex is \( (6, -2) \) and the parabola passes through the point \( (8, -1) \).

The general vertex form of a parabola is:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola. In this case, \( h = 6 \) and \( k = -2 \), so the equation becomes:
\[
y = a(x - 6)^2 - 2
\]
Now, we substitute the point \( (8, -1) \) (where \( x = 8 \) and \( y = -1 \)) into the equation to find the value of \( a \).

Substitute \( x = 8 \) and \( y = -1 \) into the equation:
\[
-1 = a(8 - 6)^2 - 2
\]
\[
-1 = a(2)^2 - 2
\]
\[
-1 = 4a - 2
\]
Add 2 to both sides:
\[
1 = 4a
\]
Solve for \( a \):
\[
a = \frac{1}{4}
\]

Now that we have \( a = \frac{1}{4} \), we can write the final equation of the parabola:
\[
y = \frac{1}{4}(x - 6)^2 - 2
\]

Would you like further details on how the calculation was done or have any specific questions?

Here are five related questions you might find helpful:
1. How do you convert a parabola from standard form to vertex form?
2. What are the steps to find the vertex of a parabola given in standard form?
3. How does changing the value of \( a \) affect the shape of a parabola?
4. Can you write the equation of a parabola given its focus and directrix?
5. How do you determine the axis of symmetry for a parabola?

**Tip**: The value of \( a \) determines whether the parabola opens upwards or downwards and how "wide" or "narrow" it is. If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.

What is the equation of a parabola with a vertex of (6, -2) that passes through (8, -1) written in vertex form?

Learn how to find the equation of a parabola with a vertex at (6, -2) that passes through the point (8, -1). This tutorial covers the vertex form of a quadratic function and how to determine the 'a' value using given points.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation