write the equation of the parabola that has it's x intercept at (1,0) and (-3,0) and it's y intercept at (0,-6)

y=?

5 answers

To start, we know that the parabola must be symmetric about the y-axis since it has x-intercepts. This means the equation will be in the form of:

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

where (h,k) is the vertex of the parabola. We can also use the y-intercept to find the "a" value.

Substitute (0,-6) for (x,y) in the equation:

-6 = a(0 - h)^2 + k

Simplify this to:

k = -6

Now we have:

y = a(x - h)^2 - 6

To find "a" and the vertex (h,k), we can use one of the x-intercepts. Let's use (1,0).

0 = a(1 - h)^2 - 6

Simplify and solve for "a":

a = 6/(1 - h)^2

Now we can substitute this back into the equation and simplify to:

y = (6/(1 - h)^2)(x - h)^2 - 6

To find "h", we can use the other x-intercept, (-3,0):

0 = (6/(1 - h)^2)(-3 - h)^2 - 6

Simplify and solve for "h":

h = -1

Substitute this back into the equation and simplify to get the final equation of the parabola:

y = (3/2)(x + 1)^2 - 6
wrong
I apologize for the error. Here is the correct solution:

Since the x-intercepts are symmetrical about the y-axis, the vertex must lie on the y-axis. Therefore, the equation of the parabola is of the form:

x = ay^2 + k

where (0, -6) is a point on the parabola. Substituting this point into the equation, we have:

0 = a(-6)^2 + k

k = 216a

Re-arranging the equation in terms of y^2, we have:

y^2 = (x - k)/a

Substituting (1, 0) and (-3, 0) into the equation of the parabola, we have:

1 = a(0)^2 + k

k = 1

-3 = a(0)^2 + k

k = -3

Solving for a, we have:

216a = 1 -> a = 1/216

Substituting this value of a and k into the equation of the parabola, we have:

y^2 = 216(x - 1)

Therefore, the equation of the parabola is y = ±√(216(x - 1)).
please give simpler
Sorry for the confusion earlier. Here's another solution.

Since the x-intercepts are at (1,0) and (-3,0), we can write the equation of the parabola as:

y = a(x-1)(x+3)

To find the value of 'a', we use the y-intercept of the parabola which is (0,-6).

Substitute x=0 and y=-6 in the above equation, we get:

-6 = a(0-1)(0+3)

-6 = -3a

a = 2

Substitute this value of 'a' in the equation of the parabola, we get:

y = 2(x-1)(x+3)

Simplifying this equation further we have:

y = 2(x^2 + 2x - 3)

y = 2x^2 + 4x - 6

Therefore, the equation of the required parabola is:

y = 2x^2 + 4x - 6.