Calculator Write and Equation for The Following Parabola

This guide explains how to write the equation of a parabola given specific points or conditions. We'll cover the standard, vertex, and intercept forms of parabola equations, provide step-by-step instructions, and include practical examples.

How to Write a Parabola Equation

Writing the equation of a parabola involves determining its shape, position, and orientation based on given information. Here's a step-by-step process:

Step 1: Identify the Given Information

You'll need at least one of the following to write a parabola equation:

Vertex and a point on the parabola
Vertex and the direction of opening
Focus and directrix
Three points on the parabola
Intercepts and vertex

Step 2: Choose the Appropriate Form

There are three main forms of parabola equations:

Standard form: (x - h)² = 4p(y - k)
Vertex form: y = a(x - h)² + k
Intercept form: y = ax² + bx + c

Step 3: Plug in the Known Values

Use the given information to substitute values into the chosen equation form. For example, if you know the vertex (h, k) and a point (x, y) on the parabola, you can solve for the coefficient.

Step 4: Solve for the Unknown

Rearrange the equation to solve for the unknown coefficient. This may involve algebra, completing the square, or using systems of equations.

Step 5: Verify the Equation

Check that the equation satisfies all given conditions. You can test points by plugging them into the equation to ensure they satisfy it.

Remember that a parabola can open up, down, left, or right. The standard form changes slightly depending on the direction of opening.

Different Forms of Parabola Equations

There are three primary forms of parabola equations, each useful in different situations:

1. Standard Form

The standard form of a parabola equation is:

(x - h)² = 4p(y - k)

Where:

(h, k) is the vertex of the parabola
p is the distance from the vertex to the focus
The parabola opens upwards if p is positive, and downwards if p is negative

2. Vertex Form

The vertex form of a parabola equation is:

y = a(x - h)² + k

Where:

(h, k) is the vertex of the parabola
a determines the parabola's width and direction (opens upwards if a is positive, downwards if negative)

3. Intercept Form

The intercept form of a parabola equation is:

y = ax² + bx + c

Where:

a, b, and c are coefficients that determine the parabola's shape and position
This form is useful when you know the y-intercepts and other points

To convert between forms, you can use algebraic manipulation and completing the square techniques.

Example Calculations

Let's look at some practical examples of writing parabola equations.

Example 1: Vertex and Point Given

Find the equation of a parabola with vertex at (2, 3) and passing through the point (4, 7).

Using the vertex form:

y = a(x - 2)² + 3

Plugging in the point (4, 7):

7 = a(4 - 2)² + 3 → 7 = 4a + 3 → 4a = 4 → a = 1

Final equation:

y = (x - 2)² + 3

Example 2: Three Points Given

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

Using the intercept form:

y = ax² + bx + c

Plugging in the points:

1 = a(1)² + b(1) + c → a + b + c = 1
2 = a(2)² + b(2) + c → 4a + 2b + c = 2
5 = a(3)² + b(3) + c → 9a + 3b + c = 5

Solving the system of equations:

Subtract first from second: 3a + b = 1
Subtract second from third: 5a + b = 3
Subtract these two results: 2a = 2 → a = 1
Then b = -2, and c = 2

Final equation:

y = x² - 2x + 2

Frequently Asked Questions

What is the difference between the standard and vertex forms of a parabola equation?

The standard form (x - h)² = 4p(y - k) emphasizes the vertex and focus, while the vertex form y = a(x - h)² + k emphasizes the vertex and the coefficient that determines the parabola's width and direction.

How do I know which form to use when writing a parabola equation?

Choose the form that matches the information you have. If you know the vertex and a point, use the vertex form. If you know the vertex and focus, use the standard form. If you have three points, use the intercept form and solve the system of equations.

Can a parabola have more than one equation?

Yes, a parabola can be represented by multiple equations, but they should all be equivalent. You can convert between forms using algebraic manipulation and completing the square techniques.

What if I only have two points on a parabola?

With just two points, you can't uniquely determine a parabola equation. You would need additional information such as the vertex or the direction of opening.