https://file.aiquickdraw.com/mathbot/file/1726076586398hem5qk2a.jpg

We are tasked with finding the equation of a parabola in vertex form. The parabola passes through points \((6, 35)\) and \((-8, 35)\), and has the vertex at \((-1, 14)\).

The general vertex form of a parabola is:

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

where \((h, k)\) is the vertex. In this case, the vertex is \((-1, 14)\), so the equation becomes:

\[
y = a(x + 1)^2 + 14
\]

### Step 1: Using a point to find \(a\)
Now, we can use one of the given points, \((6, 35)\), to solve for \(a\). Plugging \(x = 6\) and \(y = 35\) into the equation:

\[
35 = a(6 + 1)^2 + 14
\]
\[
35 = a(7)^2 + 14
\]
\[
35 = 49a + 14
\]
\[
35 - 14 = 49a
\]
\[
21 = 49a
\]
\[
a = \frac{21}{49} = 0.43
\]

### Step 2: Final Equation
Now that we have \(a = 0.43\), the equation of the parabola is:

\[
y = 0.43(x + 1)^2 + 14
\]

Thus, the correct answer from the options provided is:

\[
y = 0.43(x + 1)^2 + 14
\]

Let me know if you'd like further details or have any questions! Here are some follow-up questions to explore further:

1. How would you find the equation if the vertex were different?
2. Can you derive the axis of symmetry from the vertex form?
3. What changes if the parabola opens downward?
4. How would you convert this equation into standard form?
5. What is the importance of the coefficient \(a\) in determining the shape of the parabola?

**Tip**: The vertex form of a parabola is useful because it immediately gives you the vertex, making it easier to graph and analyze.

Write an equation of the parabola in vertex form that passes through points (6, 35), (-8, 35), and has vertex (-1, 14).

Learn how to write the equation of a parabola in vertex form that passes through points (6, 35), (-8, 35), and has the vertex (-1, 14). Includes step-by-step instructions on solving for 'a' and forming the quadratic equation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation