Question

Please look at both of these pictures it is all one problem.

Answer 1

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:

`\begin{gathered} -5=A(1)^2+B(1)+C \\ A+B+C=-5 \end{gathered}`
`\begin{gathered} -3=A(2)^2+B(2)+C \\ 4A+2B+C=-3 \end{gathered}`
`\begin{gathered} 3=A(3)^2+B(3)+C \\ 9A+3B+C=3 \end{gathered}`

We will find the value of C from the first equation and then replace it in the second and third equation:

`\begin{gathered} A+B+C=-5 \\ C=-5-A-B \end{gathered}`
`\begin{gathered} 4A+2B+C=-3 \\ 4A+2B-5-A-B=-3 \\ 3A+B=-3+5 \\ 3A+B=2 \end{gathered}`
`\begin{gathered} 9A+3B+C=3 \\ 9A+3B-5-A-B=3 \\ 8A+2B=3+5 \\ 8A+2B=8 \\ 4A+B=4 \end{gathered}`

We now use the two new equations as a subsystem of equations to find A and B:

`\begin{gathered} (4A+B)-(3A+B)=4-2 \\ 4A-3A+B-B=2 \\ A=2 \end{gathered}`
`\begin{gathered} 4(2)+B=4 \\ 8+B=4 \\ B=4-8 \\ B=-4 \end{gathered}`

With the values of A and B, we can find C as:

`\begin{gathered} C=-5-A-B \\ C=-5-2-(-4) \\ C=-7+4 \\ C=-3 \end{gathered}`

Then, the coefficients are A = 2, B = -4 and C = -3.

B) We can find the vertex of the parabola as:

`\begin{gathered} x_v=(-B)/(2A)=(-(-4))/(2*2)=(4)/(4)=1 \\ y_v=f(1)=-5 \end{gathered}`

The vertex is already one of the values in the table.

The y-intercept is given by the value of C:

`y(0)=2*0^2-4*0-3=-3`

We can find the roots (or the x-intercepts) using the quadratic equation

`\begin{gathered} x=(-B\pm√(B^2-4AC))/(2A) \\ x=(-(-4)\pm√((-4)^2-4*2*(-3)))/(2*2) \\ x=(4\pm√(16+24))/(4) \\ x=4\pm√(40) \\ x=(4\pm2√(10))/(4) \\ x=1\pm(√(10))/(2) \\ \Rightarrow x_1=1-(√(10))/(2)\approx-0.581 \\ \Rightarrow x_2=1+(√(10))/(2)\approx2.581 \end{gathered}`

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