https://file.aiquickdraw.com/mathbot/file/1726900023452y4itmmha.jpg

The image you uploaded shows a graph of a parabola, and we are tasked with finding its equation in vertex form. From the graph, we observe the following points:

- The vertex is \( (3, 150) \).
- Another point on the parabola is \( (1, 86) \).

### Step 1: Vertex Form of a Parabola
The general form of a parabola in vertex form is:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex.

From the graph:
- The vertex is \( (h, k) = (3, 150) \).
So, the equation becomes:
\[
y = a(x - 3)^2 + 150
\]

### Step 2: Solve for \( a \)
To determine \( a \), we can use the other point given on the parabola, which is \( (1, 86) \). Substituting \( x = 1 \) and \( y = 86 \) into the equation:
\[
86 = a(1 - 3)^2 + 150
\]
Simplifying:
\[
86 = a(-2)^2 + 150
\]
\[
86 = 4a + 150
\]
\[
4a = 86 - 150
\]
\[
4a = -64
\]
\[
a = -16
\]

### Step 3: Write the Equation
Now that we have \( a = -16 \), the equation of the parabola in vertex form is:
\[
y = -16(x - 3)^2 + 150
\]

This is the required equation of the parabola in vertex form.

Would you like further details or have any questions?

Here are five related questions for practice:
1. How can we convert this vertex form to standard form?
2. How would the equation change if the vertex were at \( (2, 100) \)?
3. What does the value of \( a \) represent in terms of the parabola’s shape?
4. How can you find the focus and directrix of the parabola from this equation?
5. How would the equation change if the parabola opened upwards instead of downwards?

**Tip**: The value of \( a \) determines the width and direction of the parabola. A negative \( a \) makes the parabola open downwards.

Write an equation of the parabola in vertex form given the points (3, 150) and (1, 86).

Find the equation of a parabola in vertex form using the points (3, 150) and (1, 86). This guide explains how to solve for the parameter 'a' and form the equation y = -16(x - 3)^2 + 150, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation