Chapter 2.8, Problem 23E

To determine

To state if the statement is true or false

Answer to Problem 23E

True

Explanation of Solution

Given information:

A statement saying “The graph of a quadratic model with a positive leading coefficient will have a minimum value at its vertex”.

A mathematical model represented by quadratic equation $y=a{x}^2+bx+c$ is a quadratic model.

The, leading coefficient of the quadratic equation $y=a{x}^2+bx+c$ is $a$ .

The maximum value of a function is a point where a function reaches its highest point on the graph.

Given that, the quadratic model has a positive leading coefficient.

When a graph is drawn with above conditions of quadratic equation, there will be two cases.

Case (i):

  

The above figure shows that the parabola opened rightwards whose equation is ${(y-h)}^2=4a(x-k)$ where $h=0$ and $k=0$ .

The minimum point of the parabola is at the point $(h,k)$ which is the vertex (Here, $(0,0)$ ).

Even if the value of h and k increases, the minimum point lies on the $xy$ plane.

Case (ii):

  

The above figure shows that the parabola is opened upwards whose equation is ${(x-h)}^2=4a(x-k)$ where $h=0$ and $k=0$ .

The minimum point of the parabola is at the point $(h,k)$ which is the vertex (Here, $(0,0)$ ).

Even if the value of h and k increases, the minimum point lies on the $xy$ plane.

So, the graph of a quadratic model with positive leading coefficient will have a minimum value at its vertex.