The general form of a parabola is x2−6x−2y+7=0.

What is the standard form of this parabola?

To find the standard form of the given parabola equation, we will start with the original equation and rearrange terms by completing the square.

The equation is:

\[
x^2 - 6x - 2y + 7 = 0
\]

### Step 1: Move the \(y\)-term and constant to the other side

\[
x^2 - 6x = 2y - 7
\]

### Step 2: Complete the square for the \(x\)-terms

1. Take the coefficient of \(x\) (which is \(-6\)), divide by 2, and square it:  
   \[
   \left(\frac{-6}{2}\right)^2 = (-3)^2 = 9
   \]

2. Add 9 to both sides of the equation to balance it:

   \[
   x^2 - 6x + 9 = 2y - 7 + 9
   \]

3. Simplify:

   \[
   (x - 3)^2 = 2y + 2
   \]

### Step 3: Rewrite in standard form

Now, isolate \(y\):

\[
(y + 1) = \frac{1}{2}(x - 3)^2
\]

This is the standard form of a parabola opening vertically:

\[
(y + 1) = \frac{1}{2}(x - 3)^2
\]

### Summary

The standard form is:

\[
(y + 1) = \frac{1}{2}(x - 3)^2
\]

### Characteristics of the Parabola
From this form:
- The vertex is at \((3, -1)\).
- Since the coefficient of \((x - 3)^2\) is positive, the parabola opens upwards.

---

Would you like further details, or do you have any questions?

### Relative Questions:
1. How do we determine if a parabola opens up or down from the standard form?
2. What is the significance of the vertex in a parabola?
3. How does completing the square help in transforming equations?
4. What changes if the parabola opens horizontally instead of vertically?
5. Can we graph this parabola using its standard form?

### Tip:
Completing the square is a useful technique for rewriting quadratic equations into a form that reveals the graph's features, such as the vertex and axis of symmetry.

The general form of a parabola is x^2 − 6x − 2y + 7 = 0. What is the standard form of this parabola?

Explore how to convert the general form of a parabola equation x^2 − 6x − 2y + 7 = 0 to its standard form. This problem demonstrates completing the square and provides insights into the vertex form and properties 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