a) We have to find the parameters of the parabola (A, B and C) using three points as a system of linear equations.
With the points given we can write three equations:
[tex]\begin{gathered} -5=A(1)^2+B(1)+C \\ A+B+C=-5 \end{gathered}[/tex][tex]\begin{gathered} -3=A(2)^2+B(2)+C \\ 4A+2B+C=-3 \end{gathered}[/tex][tex]\begin{gathered} 3=A(3)^2+B(3)+C \\ 9A+3B+C=3 \end{gathered}[/tex]
We will find the value of C from the first equation and then replace it in the second and third equation:
[tex]\begin{gathered} A+B+C=-5 \\ C=-5-A-B \end{gathered}[/tex][tex]\begin{gathered} 4A+2B+C=-3 \\ 4A+2B-5-A-B=-3 \\ 3A+B=-3+5 \\ 3A+B=2 \end{gathered}[/tex][tex]\begin{gathered} 9A+3B+C=3 \\ 9A+3B-5-A-B=3 \\ 8A+2B=3+5 \\ 8A+2B=8 \\ 4A+B=4 \end{gathered}[/tex]
We now use the two new equations as a subsystem of equations to find A and B:
[tex]\begin{gathered} (4A+B)-(3A+B)=4-2 \\ 4A-3A+B-B=2 \\ A=2 \end{gathered}[/tex][tex]\begin{gathered} 4(2)+B=4 \\ 8+B=4 \\ B=4-8 \\ B=-4 \end{gathered}[/tex]
With the values of A and B, we can find C as:
[tex]\begin{gathered} C=-5-A-B \\ C=-5-2-(-4) \\ C=-7+4 \\ C=-3 \end{gathered}[/tex]
Then, the coefficients are A = 2, B = -4 and C = -3.
B) We can find the vertex of the parabola as:
[tex]\begin{gathered} x_v=\frac{-B}{2A}=\frac{-(-4)}{2*2}=\frac{4}{4}=1 \\ y_v=f(1)=-5 \end{gathered}[/tex]
The vertex is already one of the values in the table.
The y-intercept is given by the value of C:
[tex]y(0)=2*0^2-4*0-3=-3[/tex]
We can find the roots (or the x-intercepts) using the quadratic equation
[tex]\begin{gathered} x=\frac{-B\pm\sqrt{B^2-4AC}}{2A} \\ x=\frac{-(-4)\pm\sqrt{(-4)^2-4*2*(-3)}}{2*2} \\ x=\frac{4\pm\sqrt{16+24}}{4} \\ x=4\pm\sqrt{40} \\ x=\frac{4\pm2\sqrt{10}}{4} \\ x=1\pm\frac{\sqrt{10}}{2} \\ \Rightarrow x_1=1-\frac{\sqrt{10}}{2}\approx-0.581 \\ \Rightarrow x_2=1+\frac{\sqrt{10}}{2}\approx2.581 \end{gathered}[/tex]
The two interceps happen at approximately -0.581 and 2.581.
Knowing this points and the points from the table we can graph the parabola as:
C) In this case, we will have a minimum profit, as the parabola opens upward and has a minimum at its vertex.
This minimum is y = -5, which represents a negative profit of 5,000.
Answer:
A) A = 2, B = -4, C = -3
B) Graph
C) We have a minimum profit that is -$5,000
