Question content area top
Part 1
Write the equation of the parabola in vertex form.
vertex left parenthesis 2 comma 1 right parenthesis​, point left parenthesis 1 comma negative 3 right parenthesis
Question content area bottom
Part 1
f left parenthesis x right parenthesisequals

enter your response here
f left parenthesis x right parenthesis
Help me solve this
View an example
Get more help

To find the equation of the parabola in vertex form, we follow these steps:

### Vertex Form of a Parabola:
The vertex form of a parabola is given by:

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

where \((h, k)\) is the vertex of the parabola, and \(a\) is a constant that determines the parabola's shape.

### Given:
- Vertex \((h, k) = (2, 1)\)
- Point \((x, y) = (1, -3)\)

### Step 1: Plug the vertex into the vertex form equation
Using the vertex \((h, k) = (2, 1)\), the equation becomes:

\[
f(x) = a(x - 2)^2 + 1
\]

### Step 2: Substitute the point \((x, y) = (1, -3)\) to find \(a\)
Substitute \(x = 1\) and \(y = -3\) into the equation:

\[
-3 = a(1 - 2)^2 + 1
\]

\[
-3 = a(-1)^2 + 1
\]

\[
-3 = a(1) + 1
\]

\[
-3 = a + 1
\]

### Step 3: Solve for \(a\)
\[
a = -3 - 1 = -4
\]

### Step 4: Write the final equation
Substitute \(a = -4\) back into the equation:

\[
f(x) = -4(x - 2)^2 + 1
\]

So, the equation of the parabola in vertex form is:

\[
f(x) = -4(x - 2)^2 + 1
\]

Would you like more details or have any other questions?

### Related Questions:
1. How do you find the vertex of a parabola given its equation in standard form?
2. What does the value of \(a\) represent in the vertex form of a parabola?
3. How can you convert the vertex form of a parabola to standard form?
4. What are the effects of changing \(a\) on the shape of the parabola?
5. How do you find the x-intercepts and y-intercept of a parabola in vertex form?

### Tip:
When solving problems involving parabolas, always ensure to identify the vertex correctly, as it's crucial for setting up the vertex form equation accurately.

Learn how to write the equation of a parabola in vertex form when given the vertex (2, 1) and a point (1, -3). This detailed solution covers the vertex form formula and step-by-step calculations to find the constant 'a'. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation