Find the equation of the parabola with its axis parallel to x-axis and passing through the points (-2,1),(1,2),(-1,3)

x = ay^2 + by + c

just plug in your points.

a+b+c = -2
and so on. Then solve for a,b,c

(_4,2)(-6,4),(-12,6)find equation of parabola

To find the equation of a parabola with its axis parallel to the x-axis, we need to 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.

To find the vertex, we can use the fact that the axis of symmetry of the parabola is parallel to the y-axis and passes through the midpoint of the directrix. The midpoint of the directrix can be found by taking the average of the y-coordinates of the given points.

Midpoint of the directrix = (1 + 2 + 3) / 3 = 2

Therefore, the vertex should also have a y-coordinate of 2.

Let's substitute the vertex coordinates (h = -2, k = 2) and choose one of the given points to find the value of a.

Using the point (1, 2):

2 = a(1 - (-2))^2 + 2
2 = a(3)^2 + 2
2 = 9a + 2
9a = 0
a = 0

Finally, substitute the value of a into the equation:

y = 0(x - (-2))^2 + 2
y = 0(x + 2)^2 + 2

Therefore, the equation of the parabola with its axis parallel to the x-axis and passing through the points (-2,1), (1,2), (-1,3) is:

y = 0(x + 2)^2 + 2

To find the equation of the parabola with its axis parallel to the x-axis and passing through the given points, we can use the vertex form of the equation:

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

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

Step 1: Find the vertex (h, k)
Since the axis of the parabola is parallel to the x-axis, the vertex will have the form (h, k).

To find the vertex, we can take the average of the x-coordinates of the given points to get the value of h, and the average of the y-coordinates to get the value of k.

In this case, the x-coordinates are -2, 1, and -1, so:
h = (-2 + 1 - 1)/3 = -2/3

The y-coordinates are 1, 2, and 3, so:
k = (1 + 2 + 3)/3 = 2

Therefore, the vertex is V(-2/3, 2).

Step 2: Substitute the vertex into the equation
Using the vertex form of the equation, we can substitute the values of h and k into the equation to get the final equation of the parabola.

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

Step 3: Use one of the given points to find the value of a
We still need to determine the value of a to complete the equation. For this, we can use one of the given points on the parabola. Let's use the point (1, 2).

Substituting the values (x,y) = (1, 2) into the equation, we have:
2 = a(1 + 2/3)^2 + 2
2 = a(9/9) + 2
2 = a + 2
a = 2 - 2
a = 0

Therefore, the value of a is 0.

Step 4: Write the final equation
Substituting the value of a = 0 into the equation, we get:
y = 0(x + 2/3)^2 + 2

Simplifying further, we have:
y = 0 + 2
y = 2

So, the equation of the parabola with its axis parallel to the x-axis and passing through the points (-2, 1), (1, 2), and (-1, 3) is:
y = 2