Question

Write the equation of the parabola that has its x-intercepts at (-2-√2, 0) and (-2√2, 0) and passes through the point (-1, 1)?

Answer 1

The parabola with x-intercepts at (-2-√2, 0) and (-2+√2, 0) and passing through the point (-1, 1) has the equation y = -x^2 - 4x - 2.

Step-by-step explanation:

To find the equation of a parabola that has its x-intercepts at (-2-√2, 0) and (-2+√2, 0) and passes through the point (-1, 1), we start by using the fact that the x-intercepts represent the roots of the quadratic equation. From this, the quadratic equation in factorized form is given by:

`f(x) = a(x - (-2 - √2))(x - (-2 + √2))`

Expanding the factors, we get:

f(x) = a(x + 2 + √2)(x + 2 - √2)

Which simplifies to:

f(x) = a((x + 2)^2 - (√2)^2)

f(x) = a(x^2 + 4x + 4 - 2)

f(x) = a(x^2 + 4x + 2)

Since the parabola also passes through the point (-1, 1), we substitute x with -1 and y with 1 to find the value of a:
1 = a((-1)^2 + 4(-1) + 2)
1 = a(1 - 4 + 2)
1 = a(-1)

This gives us a = -1. Therefore, the equation of the parabola is:

y = -1(x^2 + 4x + 2)

Or simplified to:

y = -x^2 - 4x - 2

Answer 2

The equation of the parabola is y = (1/(1 + 8√2))(x + 2√2)² + (1 - 1/(1 + 8√2))(x + 2√2)

Step-by-step explanation:

The equation of the parabola can be found using the vertex form: y = a(x-h)² + k, where (h,k) is the vertex. Given that the parabola has its x-intercepts at (-2-√2, 0) and (-2√2, 0), we can determine the vertex. Since the x-coordinate of the vertex is the average of the x-intercepts, h = (-2-√2 + -2√2)/2 = -√2 - √2 = -2√2. Plugging in the coordinates of the point (-1, 1), we get 1 = a(-1 - (-2√2))² + k. Simplifying, we have a(1 + 2√2)² + k = 1.

Additionally, since the point (-1, 1) lies on the parabola, substituting the coordinates into the equation gives us 1 = a(-1 - (-2√2))² + k. Simplifying, we have a(1 + 2√2)² + k = 1.

Now, we have two equations:

a(1 + 2√2)² + k = 1

a(-2√2)² + k = 0

Solving these equations simultaneously, we find a = 1/(1 + 8√2) and k = 1 - a(1 + 2√2)². Substituting the values of a and k into the vertex form equation, we get the equation of the parabola:

y = (1/(1 + 8√2))(x + 2√2)² + (1 - 1/(1 + 8√2))(x + 2√2)