a parabola whose axis is parallel to the y axis passess through the points {1,1), (2.2) and (-1,5). Find the equation of the parabola
let the equation have the form
y = ax^2 + bx + c
for (1,1) --> 1 = a + b + c
for (2,2) --> 2 = 4a + 2b + c
for (-1,5) --? 5 = a - b + c
subtract #1 - #3: 2b = -4 , b = -2
subtract #2 - #1: 3a + b = 1
sub in b = -2
3a - 2 = 1
3a = 3
a = 1
sub into the first:
1 - 2 + c = 1
c = 2
your equation is:
y = x^2 - 2x + 2
To find the equation of a parabola given three points, we can use the standard form of a parabola equation, which is:
y = a * x^2 + b * x + c
where 'a', 'b', and 'c' are constants that need to be determined.
Let's start by substituting the x and y values of the first point (1, 1) into the equation:
1 = a * (1)^2 + b * (1) + c
1 = a + b + c ... equation (1)
Next, we substitute the x and y values of the second point (2, 2):
2 = a * (2)^2 + b * (2) + c
2 = 4a + 2b + c ... equation (2)
Finally, we substitute the x and y values of the third point (-1, 5):
5 = a * (-1)^2 + b * (-1) + c
5 = a - b + c ... equation (3)
Now, we have a system of three equations (equations 1, 2, and 3) with three variables (a, b, and c).
To solve this system of equations, we can use substitution, elimination, or matrix methods. In this case, let's use elimination to eliminate variable 'c':
Subtract equation (2) from equation (1):
1 - 2 = (a + b + c) - (4a + 2b + c)
-1 = -3a - b
Next, subtract equation (1) from equation (3):
5 - 1 = (a - b + c) - (a + b + c)
4 = -2b
Now, we have two equations:
-1 = -3a - b ... equation (4)
4 = -2b ... equation (5)
Solving equation (5) for 'b':
b = -2/(-2)
b = 2
Substitute the value of 'b' into equation (4):
-1 = -3a - 2
3a = -1 + 2
3a = 1
a = 1/3
Now, we have the values of 'a' and 'b'. Substitute them into the standard form equation:
y = (1/3) * x^2 + 2 * x + c
To find the value of 'c', substitute the x and y values of any of the given points. Let's use the first point (1, 1):
1 = (1/3) * (1)^2 + 2 * (1) + c
1 = 1/3 + 2 + c
1 = 7/3 + c
Simplify the equation:
1 - 7/3 = c
(3 - 7)/3 = c
-4/3 = c
Finally, substitute the value of 'c' into the equation:
y = (1/3) * x^2 + 2 * x - 4/3
Therefore, the equation of the parabola is y = (1/3) * x^2 + 2 * x - 4/3.