Answer:
The equation of the parabola is y = (1/16)(x + 1)^2
Explanation:
To find the equation of a parabola given its intercepts and a point it passes through, we can use the standard form of the equation for a parabola:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola and a is a constant that determines the shape of the parabola.
Given the intercepts (1, 0) and (-3, 0), we know that the parabola crosses the x-axis at these points.
This means that (1, 0) and (-3, 0) are the x-intercepts, or roots, of the equation.
We can set up two equations using the x-intercepts:
When x = 1, y = 0: 0 = a(1 - h)^2 + k
When x = -3, y = 0: 0 = a(-3 - h)^2 + k
We are also given a point that the parabola passes through, (-1, -10).
Plugging in these coordinates into the equation of the parabola, we get:
-10 = a(-1 - h)^2 + k
Now we have a system of three equations:
0 = a(1 - h)^2 + k
0 = a(-3 - h)^2 + k
-10 = a(-1 - h)^2 + k
We can solve this system of equations to find the values of a, h, and k, and then substitute these values into the equation of the parabola.
Solving the system of equations, we find:
a = 1/16
h = -1
k = 0
Substituting these values into the equation of the parabola, we get:
y = (1/16)(x + 1)^2 + 0
Simplifying, the equation of the parabola is:
y = (1/16)(x + 1)^2
Thus,
y = (1/16)(x + 1)^2 is the equation of the parabola.