Question

Write the equation of the parabola that has its x intercepts at (1 sqrt5, 0), (1-sqrt5, 0) and passes thru the point (4,8)

Answer 1

The equation of the parabola is y = (2/(3 + √5))(x - (1 + √5))(x - (1 - √5)).

Step-by-step explanation:

The equation of the parabola can be written in the form y = ax² + bx + c. To find the equation, we need to use the x-intercepts and the point that the parabola passes through.

Vertex: Since the parabola has its x-intercepts at equidistant points from the vertex, the vertex lies exactly between them. Therefore, the vertex has coordinates (1, 0).

Focal length: We know the distance from the vertex to the focus is equal to the distance from the vertex to the x-intercept. In this case, that distance is 1.

Form of the equation: Knowing the vertex and focal length, we can use the standard equation of a parabola with a vertical axis:

Since the x-intercepts are given as (1 + √5, 0) and (1 - √5, 0), we can write the equation in factored form as y = a(x - (1 + √5))(x - (1 - √5)).

Plugging in the coordinates of the point (4,8), we get 8 = a(4 - (1 + √5))(4 - (1 - √5)). Solving for a, we find a = 2/(3 + √5).

Therefore, the equation of the parabola is y = (2/(3 + √5))(x - (1 + √5))(x - (1 - √5)).

Answer 2

The equation of the parabola is determined by its x-intercepts and a given point. Using the intercepts and the point (4,8), we can find the value of 'a' in the parabola's standard form equation.

Step-by-step explanation:

The equation of a parabola with given x-intercepts can be found using the form y = a(x - p)(x - q), where p and q are the x-intercepts of the parabola. In this case, the x-intercepts are given as (1 + √5, 0) and (1 - √5, 0), so we can start with y = a(x - (1 + √5))(x - (1 - √5)). To find the value of a, we use the point that the parabola passes through, which is (4, 8). Plugging these values into the equation gives:

8 = a(4 - (1 + √5))(4 - (1 - √5))

Then, we solve for a and get the final equation. The quadratic formula and other related content provided are not directly relevant to writing the equation of a parabola based on given intercepts and a point it passes through.