the following table shows the revenue for a company generates based on the increases in the price of the product. What is the y-value of the Vertex of the parabola that models the date?

From the given that it is noticed that the parabolic function passing through the points (1,1045), (3,1105) and (5,1125). It means the function must be satisfied by these points.

\(1045=a(1)^2+b(1)+c\)

\(1045=a+b+c\) ....(1)

\(1105=a(3)^2+b(3)+c\)

\(1105=9a+3b+c\) ....(2)

\(1125=a(5)^2+b(5)+c\)

\(1125=25a+5b+c\) ....(3)

On solving (1), (2) and (3) we get,

\(a=-5\)

\(b=50\)

\(c=1000\)

Therefore the equation of parabola is

\(f(x)=-5x^2+50x+1000\)

The vertex of the parabola is

\((\frac{-b}{2a},f(\frac{-b}{2a}))\)

\(\frac{-b}{2a}=-\frac{50}{2(-5)}=5\)

\(f(5)=1125\)

Therefore the vertex is (5,1125) and y-value of the Vertex of the parabola that models the data is 1125.

ap8997154 ap8997154 · Answer 2 · 2022-02-01T07:24:37-05:00

The vertexes of the parabola are, (5, 1125).

Explanation

The table given to us in the problem are the data points that will lie on the parabola, therefore,

Point 1 = (1, 1045)

Point 2 = (3, 1105)

Point 3 = (5, 1125)

Point 4 = (3, 1105)

Point 5 = (1, 1045)

Equation of a Parabola,

We know that the equation of a parabola is given as,

\(y = ax^2 +bx+c\)

For point 1,

Point 1 = (1, 1045)

Substituting the value in the equation of a parabola,

\(1045 = a(1)^2 +(1)b+c\\\\1045 = a+b+c\)..... equation 1,

For point 2,

Point 2 = (3, 1105)

Substituting the value in the equation of a parabola,

\(1105 = a(3)^2 +(3)b+c\\\\1105= 9a+3b+c\)..... equation 2,

For point 3,

Point 3 = (5, 1125)

Substituting the value in the equation of a parabola,

\(1125= a(5)^2 +(5)b+c\\\\1125= 25a+5b+c\)..... equation 3,

Solving the three equations we get,

a = -5,

b = 50,

c = 1000

Substitute the values in the equation of a parabola,

\(y=f(x) = -5x^2 +50x +1000\)

How to find Vertexes of a parabola?

To find the vertex of a parabolic equation we bring the equation into the form,

\(y = a(x-h)+k\\\) , where h and k are the vertexes of the parabola.

Vertexes of the parabola

Vertex of the Parabola,

\(y=f(x) = -5x^2 +50x +1000\\\\y = -5x^2 +50x +1000\\\\y =-5(x^2 -10x)+1000\\\\y =-5(x^2 -10x+25-25)+1000\\\\y =-5(x^2 -10x+25)+ (-5\times -25)+1000\\\\y =-5(x^2 -10x +25)+125+1000\\\\y =-5(x^2 -5)^2+1125\)

Comparing it to the equation, \(y = a(x-h)+k\\\),

the vertexes of the parabola are,

(5, 1125)

Learn more about the Equation of a Parabola:

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