Solution:
Using the equation:
[tex]y=ax^2+bx+c[/tex]
Given the table of values of x and y as shown below:
From the table, when x equals zero, we have y to be -1.4.
Substitute these values of x and y into the above equation.
Thus, we have
[tex]\begin{gathered} y=ax^2+bx+c \\ \Rightarrow-1.4=a(0)^2+b(0)+c \\ \therefore c=-1.4 \end{gathered}[/tex]
Substitute the obtained value of c into the equation.
This gives
[tex]y=ax^2+bx-1.4\text{ ---- equation 1}[/tex]
Also, when x equals 0.905, we have y to be zero.
thus, we similarly substitute the values of x and y into the equation.
[tex]\begin{gathered} y=ax^2+bx+c \\ \Rightarrow0=a(0.905)^2+b(0.905)+c \\ 0=0.819025a+0.905b+c \\ \text{but c=-1.4} \\ \text{thus, we have} \\ 0=0.819025a+0.905b-1.4 \\ \Rightarrow0.819025a+0.905b=1.4\text{ ----- equation 2} \end{gathered}[/tex]
When x equals 2, y equals 0.6. substitiute the values of x and y into the equation.
[tex]\begin{gathered} y=ax^2+bx+c \\ \Rightarrow0.6=4a+2b+\text{c} \\ \text{where c=-1.4} \\ \text{thus,} \\ 0.6=4a+2b-1.4 \\ \text{add 1.4 to both sides of the equation,} \\ 0.6+1.4=4a+2b-1.4+1.4 \\ \Rightarrow4a+2b=2\text{ } \\ \text{divide through by 2 reduces the equation to be} \\ 2a+b=1\text{ ------ equation 3} \end{gathered}[/tex]
From equation 3, make b the subject of the equation or formula.
Thus,
[tex]\begin{gathered} 2a+b=1 \\ \Rightarrow b=1-2a\text{ ---- equation 4} \end{gathered}[/tex]
substitute equation 4 intp equation 2.
Thus,
[tex]\begin{gathered} 0.819025a+0.905b=1.4 \\ \text{where} \\ b=1-2a \\ \text{thus,} \\ 0.819025a+0.905(1-2a)=1.4 \\ open\text{ parentheses} \\ 0.819025a+0.905-1.81a=1.4 \\ \text{collect like terms} \\ -0.990975a+0.905=1.4 \\ \text{subtract 0.905 from both sides of the equation} \\ -0.990975a+0.905-0.905=1.4-0.905 \\ \Rightarrow-0.990975a=0.495 \\ \text{divide both sides by the coefficient of a} \\ \frac{-0.990975a}{-0.990975}=\frac{0.495}{-0.990975} \\ \Rightarrow a=-0.49950 \end{gathered}[/tex]
To solve for b, substitute the obtained value of a into equation 4.
From equation 4,
[tex]\begin{gathered} b=1-2a \\ \text{where} \\ a=-0.49950 \\ \text{thus,} \\ b=1-2(-0.49950) \\ \Rightarrow b=1.999 \end{gathered}[/tex]
From the obtained values of a, b, and c, equation 1 becomes
[tex]\begin{gathered} y=ax^2+bx-1.4\text{ } \\ \Rightarrow y=-0.49950x^2+1.999x-1.4 \end{gathered}[/tex]
Hence, the equation of the parabola is
[tex]y=-0.49950x^2+1.999x-1.4[/tex]
