Give the equation of the parabola when y_intercept is 40,x1=4 and x2=5 ?

To find the equation of the parabola, we first need to determine the vertex of the parabola using the x-intercepts provided.

The x-intercepts are x1=4 and x2=5. The average of the x-intercepts will give us the x-coordinate of the vertex:
x-coordinate of vertex = (x1 + x2) / 2
x-coordinate of vertex = (4 + 5) / 2
x-coordinate of vertex = 4.5

Now, we can substitute the x-coordinate of the vertex and the y-intercept into the standard form of the equation of a parabola:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

Given that the y-intercept is 40, we have the point (0, 40) on the parabola. Substituting this point into the equation, we get:

40 = a(0 - 4.5)^2 + k
40 = a(20.25) + k
40 = 20.25a + k

Now, we substitute the vertex coordinates (4.5, k) into the equation:

k = 20.25a + k
40 = 20.25a + 40
0 = 20.25a
a = 0

Therefore, the equation of the parabola is y = 0(x - 4.5)^2 + 40
Simplified to y = 40

So, the equation of the parabola is y = 40.

The equation of a parabola can be written in the form y = ax^2 + bx + c, where a, b, and c are constants. We are given that the y-intercept is 40, so c = 40. We are also given two points on the parabola, (x1, y1) = (4, y1) and (x2, y2) = (5, y2). We can use these points to solve for the constants a and b.

Substituting the first point into the equation, we get:

y1 = a(x1)^2 + b(x1) + c
y1 = a(4)^2 + b(4) + 40
y1 = 16a + 4b + 40

Substituting the second point into the equation, we get:

y2 = a(x2)^2 + b(x2) + c
y2 = a(5)^2 + b(5) + 40
y2 = 25a + 5b + 40

Solving for y1 - y2, we get:

y1 - y2 = (16a + 4b + 40) - (25a + 5b + 40)
y1 - y2 = 16a - 25a + 4b - 5b
y1 - y2 = -9a - b

Substituting the given values of x1, x2, and y1 - y2, we get:

4 - y2 = -9a - b

We can use this equation to solve for b in terms of a:

b = 4 - y2 + 9a

Substituting this expression for b into the equation for y1, we get:

y1 = 16a + 4(4 - y2 + 9a) + 40
y1 = 16a + 16 - 4y2 + 36a + 40
y1 = 52a - 4y2 + 56

Now we can substitute the given value of y1 into this equation:

40 = 52a - 4y2 + 56
-16 = 52a - 4y2
-4 = 13a - y2

Finally, we can solve for a:

a = (4 + y2) / 13

Substituting this expression for a into the equation for b, we get:

b = 4 - y2 + 9((4 + y2) / 13)
b = 4 - y2 + 36 + 9y2 / 13
b = 40 + 8y2 / 13

Therefore, the equation of the parabola is:

y = ((4 + y2) / 13)x^2 + (40 + 8y2 / 13)x + 40