Write the equation of the parabola in vertex form. 
vertex left parenthesis 2 comma 1 right parenthesis(2,1)​, point left parenthesis 3 comma negative 4 right parenthesis(3,−4)

To find the equation of the parabola in vertex form, we start with the general formula for a parabola in vertex form:

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

Where \((h, k)\) is the vertex of the parabola. For this case, the vertex is given as \((2, 1)\), so the equation becomes:

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

Next, we use the additional point \((3, -4)\) to solve for the value of \(a\). Substituting \(x = 3\) and \(y = -4\) into the equation:

\[
-4 = a(3 - 2)^2 + 1
\]

Simplifying the equation:

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

\[
-4 = a + 1
\]

Now, subtract 1 from both sides:

\[
-5 = a
\]

Thus, the value of \(a\) is \(-5\). Therefore, the equation of the parabola is:

\[
y = -5(x - 2)^2 + 1
\]

This is the equation of the parabola in vertex form.

Do you have any questions or need further details? 

Here are five related questions to explore:
1. How do you find the equation of a parabola given three points?
2. What is the significance of the vertex in the graph of a parabola?
3. How do you convert a standard-form quadratic equation into vertex form?
4. How does changing the value of \(a\) affect the shape of the parabola?
5. How can you find the focus and directrix of a parabola from its vertex form?

**Tip:** The sign of \(a\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

Write the equation of the parabola in vertex form. Vertex (2,1), point (3,-4)

Learn how to find the equation of a parabola in vertex form with vertex (2,1) and a point (3,-4). This step-by-step guide covers key algebraic concepts including the vertex form equation y = a(x-h)^2 + k, solving for the parameter 'a,' and understanding the structure of parabolas. 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