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=?
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 = Β±β(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.
To write the equation of a parabola, we need to use the vertex form of the equation:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola.
Step 1: Determine the vertex of the parabola
Since the x-intercepts are given as (1, 0) and (-3, 0), we can find the average of these two x-values to get the x-coordinate of the vertex. The y-coordinate of the vertex will be the y-intercept, which is (0, -6).
Average of the x-intercepts:
x-coordinate of the vertex = (1 + (-3)) / 2 = -2 / 2 = -1
Therefore, the vertex of the parabola is (-1, -6).
Step 2: Substitute the vertex coordinates into the equation
Using the coordinates of the vertex, we have:
y = a(x - (-1))^2 - 6
Simplifying further, we get:
y = a(x + 1)^2 - 6
Step 3: Use one of the given points to find the value of 'a'
We can use either of the x-intercepts (1, 0) or (-3, 0) to find the value of 'a'.
Let's use the point (1, 0):
0 = a(1 + 1)^2 - 6
0 = 4a - 6
6 = 4a
a = 6 / 4
a = 3 / 2
Step 4: Substitute the value of 'a' back into the equation
Now that we have the value of 'a', we can substitute it back into the equation:
y = (3 / 2)(x + 1)^2 - 6
Therefore, the equation of the parabola is:
y = (3 / 2)(x + 1)^2 - 6