To find the equation of a parabola, you typically need some specific information about its shape and position. Here are some common scenarios and methods:

1. **Vertex Form**:
   The equation of a parabola can be written as \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola, and \( a \) determines the direction and width of the parabola.

   **Example**: Find the equation of a parabola with vertex at \((2, 3)\) and passing through the point \((3, 5)\).

   **Steps**:
   1. Start with the vertex form: \( y = a(x - 2)^2 + 3 \).
   2. Use the point \((3, 5)\) to find \( a \):
      \[
      5 = a(3 - 2)^2 + 3 \\
      5 = a(1)^2 + 3 \\
      5 = a + 3 \\
      a = 2
      \]
   3. The equation is \( y = 2(x - 2)^2 + 3 \).

2. **Standard Form**:
   The equation can also be written as \( y = ax^2 + bx + c \).

   **Example**: Find the equation of a parabola passing through points \((1, 2)\), \((2, 3)\), and \((3, 5)\).

   **Steps**:
   1. Use the standard form: \( y = ax^2 + bx + c \).
   2. Set up a system of equations using the given points:
      \[
      2 = a(1)^2 + b(1) + c \implies a + b + c = 2 \\
      3 = a(2)^2 + b(2) + c \implies 4a + 2b + c = 3 \\
      5 = a(3)^2 + b(3) + c \implies 9a + 3b + c = 5
      \]
   3. Solve this system of linear equations to find \( a \), \( b \), and \( c \).

Would you like more details or have any questions on these methods?

Here are some related questions to expand on this topic:

1. How do you find the vertex of a parabola given its standard form equation?
2. What is the focus and directrix of a parabola, and how do you find them?
3. How do you convert a parabola's equation from standard form to vertex form?
4. How do you determine the direction in which a parabola opens?
5. What is the axis of symmetry of a parabola, and how do you find it?
6. How do you find the x-intercepts (roots) of a parabola?
7. What is the relationship between the coefficient \( a \) and the width of the parabola?
8. How do you graph a parabola given its equation in vertex form?

**Tip:** To solve systems of linear equations, you can use methods such as substitution, elimination, or matrix operations (like Gaussian elimination).

Learn how to find the equation of a parabola using vertex form and standard form with detailed step-by-step solutions. This problem covers key concepts including parabolas, vertex form, standard form, and solving systems of equations. 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