[tex]y\text{ = -}0.4603x^2\text{ + 1.8587x -1.3}[/tex]
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:
[tex]y=ax^2\text{ + bx + c}[/tex][tex]\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}[/tex]
[tex]\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}[/tex][tex]\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}[/tex]
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
[tex]\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{ = }\frac{1.672843}{0.9} \\ b\text{ = 1}.8587 \end{gathered}[/tex]
We will substitute the values of a, b, and c in the formula. The equation of the parabola will become:
[tex]\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}[/tex]
