The standard form of a parabola with points through (2, 0) (3, 2) (4, 6)
is y = x² - 3x + 2 ⇒ 3rd answer
Step-by-step explanation:
The standard form of a parabola is y = ax² + bx + c, where a, b , c are constant
To find a , b , c
- You must have 3 points lie on the parabola
- Substitute the coordinates of each point in the equation to make system of equations of a , b and c
- Solve the system of equation to find them
∵ The standard form of a parabola is y = ax² + bx + c
∵ The parabola passes through points (2 , 0) , (3 , 2) , (4 , 6)
- Substitute the coordinates of each point in the equation
Point (2 , 0)
∵ x = 2 and y = 0
∴ 0 = a(2)² + b(2) + c
∴ 0 = 4a + 2b + c
- Switch the two sides
∴ 4a + 2b + c = 0 ⇒ (1)
Point (3 , 2)
∵ x = 3 and 2 = 0
∴ 2 = a(3)² + b(3) + c
∴ 2 = 9a + 3b + c
- Switch the two sides
∴ 9a + 3b + c = 2 ⇒ (2)
Point (4 , 6)
∵ x = 4 and y = 6
∴ 6 = a(4)² + b(4) + c
∴ 6 = 16a + 4b + c
- Switch the two sides
∴ 16a + 4b + c = 6 ⇒ (3)
Subtract equation (1) from equations (2) and (3)
∴ 5a + b = 2 ⇒ (4)
∴ 12a + 2b = 6 ⇒ (5)
- Multiply equation (4) by -2 to eliminate b
∴ -10a - 2b = -4 ⇒ (6)
- Add equations (5) and (6)
∴ 2a = 2
- Divide both sides by 2
∴ a = 1
Substitute the value of a in equation (4) to find b
∵ 5(1) + b = 2
∴ 5 + b = 2
- Subtract 5 from both sides
∴ b = -3
Substitute the value of a and b in equation (1) to find c
∵ 4(1) + 2(-3) + c = 0
∴ 4 - 6 + c = 0
- Add like terms
∴ -2 + c = 0
- Add 2 to both sides
∴ c = 2
Substitute the values of a , b , c in the standard form above
∵ y = ax² + bx + c
∵ a = 1 , b = -3 , c = 2
∴ y = (1)x² + (-3)x + (2)
∴ y = x² - 3x + 2
The standard form of a parabola with points through (2, 0) (3, 2)
(4, 6) is y = x² - 3x + 2
There is another solution you can substitute the x-coordinate of each point in each answer to find the corresponding value of y, if the value of y gives the same value of the y-coordinate of the point for the three points then this answer is the standard form of the parabola
Learn more:
You can learn more about the parabola in brainly.com/question/8054589
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