Final answer:
To find the equation of a parabola with given x-intercepts and a specific point, we form a quadratic equation y = a(x - 1 - √5)(x - 1 + √5), plug in the point to find 'a', and write the final equation. In this case, we found 'a' to be 2, and the equation of the parabola is y = 2(x^2 - 2x - 4).
Step-by-step explanation:
The equation for a parabola that has specific x-intercepts can be expressed in the form y = a(x-h)2 + k, where (h,k) is the vertex of the parabola. Since the given x-intercepts are (1+ √5 ,0) and (1√5 ,0), we can express the equation as y = a(x - (1+ √5))(x - (1√5)).
To find the value of 'a', we use the given point (4,8) that lies on the parabola, which gives us the equation 8 = a(4 - (1+ √5))(4 - (1√5)). Simplifying and solving this equation will provide us with the value of 'a'. After that, we can write the complete equation of the parabola.
First, let's expand the equation using the given x-intercepts:
y = a(x - 1 - √5)(x - 1 + √5)
y = a((x - 1)2 - (√5)2)
y = a(x2 - 2x + 1 - 5)
y = a(x2 - 2x - 4)
Now, insert the point (4,8) into the equation:
8 = a(42 - 2∙4 - 4)
8 = 16a - 8a - 4a
8 = 4
This implies that a = 2.
The final equation of the parabola can now be written as:
y = 2(x2 - 2x - 4)