Chapter 9.1, Problem 72HP

To determine

To Describe:

A real-world situation that involves a quadratic equation. Explain what the vertex represents.

Explanation of Solution

Concept Used :

• Quadratic Functions:
• Quadratic functions are non-linear and are of the form:

$f(x)=a{x}^2+bx+c;a\ne 0$

This is the standard form of Quadratic function.

• Shape of the graph of a Quadratic function is called Parabola.
• Parabolas are symmetric about a central line called Axis Of Symmetry.
• The axis of symmetry intersects a parabola only at one point , called the Vertex.
• Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves.

$x=-\frac{b}{2a}$

• y-intercept = The y coordinate of the point at which the graph intersects y-axis = c
• Graph:

  

• If a > 0 ,
• The graph of $a{x}^2+bx+c$ opens upward.
• The lowest point on the graph is the minimum.
• If a < 0 ,
• The graph of $a{x}^2+bx+c$ opens downward.
• The highest point on the graph is the maximum.
• The maximum or the minimum is the vertex.

Calculation:

We can represent the path travelled by a ball thrown in the air through a downward parabola .

So, the equation of the path of a ball thrown in the air will be a quadratic equation , where the value of a < 0 .

And vertex will be the maximum point of the parabola . So, it will represent the maximum height that the ball attains during its flight in the air.