Question

How to find a equation of a parabola with 3 points

Answer 1

`y\text{ = -}0.4603x^2\text{ + 1.8587x -1.3}`

Step-by-step explanation:

When given a parabola and we want to get its equation, we will pick three points on the parabola. To make it easy, two of the points should be the x and y -intercept

Picking three points from the parabola:

x-intercept: (0.9, 0)

y-intercept: (0, -1.3)

3rd point: (1.4, 0.4)

Next we will insert the value above into the general formula for a parabola (quadratic function).

The formula is given as:

`y=ax^2\text{ + bx + c}`
`\begin{gathered} \text{when point is (}0,\text{ -1.3): x = 0, y = -1.3} \\ y=ax^2+\text{ bx + c} \\ -1.3=a(0)^2\text{ + }b(0)\text{ + c} \\ -1.3\text{ = 0 + 0 + c} \\ -1.3\text{ = c . . .(1)} \end{gathered}`

`\begin{gathered} \text{when point is (0.9, 0): x = 0.9, y = 0} \\ y=ax^2+\text{ bx + c} \\ 0=a(0.9)^2\text{ + b(0.9) + c} \\ 0=0.81a\text{ + 0.9b + c } \\ \text{from equation (1), c = -1.3} \\ 0\text{ = 0.}81a\text{ + 0.9b + (-1.3)} \\ 0\text{ = 0.}81a\text{ + 0.9b -1.3} \\ \\ \text{0.}81a\text{ + 0.9b = 1.3 . . . (2)} \end{gathered}`
`\begin{gathered} \text{when point is (}1.4,\text{ 0.4): x = 1}.4,\text{ y = 0.4} \\ y=ax^2\text{ + bx + c} \\ 0.4=a(1.4)^2\text{ + b(1.4) + c} \\ 0.4\text{ = 1.96a + 1.4b + c} \\ \\ \text{from equation 1, c = -1.3} \\ 0.4\text{ = 1.96a + 1.4b + (-1.3)} \\ 0.4\text{ = 1.96a + 1.4b - 1.3} \\ 0.4\text{ + 1.3 = 1.96a + 1.4b } \\ 1.7\text{ = 1.96a + 1.4b } \\ \text{ 1.96a + 1.4b = 1.7 . . .(3)} \end{gathered}`

To get a and b, we will solve equation (2) and (3) simultaneously:

0.81a + 0.9b = 1.3 (2)

1.96a + 1.4b = 1.7 (3)

Using elimination method:

let's eliminate b. To do this we will multiply equation (2) by 1.4 and equation (3) by 0.9.

By so doing they will both have same coefficient in b and we will be able to do the eilmination

1.134a + 1.26b = 1.82 (2*)

1.764a + 1.26b = 1.53 (3*)

subtract equation (2*) from equation (3*):

1.764a - 1.134a + 126b - 126b = 1.53 - 1.82

0.63a = -0.29

divide both sides by 0.63:

a = -0.29/0.63

a = -0.4603

substitute for a in any of the equations:

using equation (2): 0.81a + 0.9b = 1.3

`\begin{gathered} 0.81(-0.4603)+0.9b=1.3 \\ -0.372843\text{ + 0.9b = 1.3} \\ 0.9b\text{ = 1.3 + 0.372}843 \\ 0.9b\text{ = }1.672843 \\ \text{divide both sides by 0.9:} \\ b\text{ = }(1.672843)/(0.9) \\ b\text{ = 1}.8587 \end{gathered}`

We will substitute the values of a, b, and c in the formula. The equation of the parabola will become:

`\begin{gathered} y\text{ = -}0.4603x^2\text{ + 1.8587x + (-1.3)} \\ \\ The\text{ equation of the parabola is:} \\ y\text{ = -}0.4603x^2\text{ + 1.8587x -1.3} \end{gathered}`