Chapter 9.1, Problem 70HP

To determine

To formulate an equation for the graph shown.

Answer to Problem 70HP

The quadratic equation representing the parabola shown in the graph is $y=-x^2+6x+16$

Explanation of Solution

Given information :

The given figure shows a downward facing parabola.

Vertex of the graph is at $(3,25)$

y-intercept is $(0,16)$

Hence, the axis of symmetry is $x=3$

x-intercept is $(8,0)$

Formula used :

The graph of a quadratic equation ( $y=a{x}^2+bx+c$ ) is a parabola. The parabola is symmetric in nature and the point of symmetry is known as the vertex. This vertex lies on the line of symmetry and is symmetric about this line.

The axis of symmetry bisects the parabola into two equal parts. Hence each point on the parabola would have an equal point on the other side of the axis of symmetry

This vertex point shall be:

Highest point (if $a<0$ ) and be called the maximum. Here, the graph opens downwards.

Or, lowest point (if $a>0$ ) and can be called the minimum. Here the graph opens upwards.

In this case, ‘a’ is lesser than 0 hence the graph will have a maximum and will open downwards.

A parabola always points to infinity, either negative or positive.

Formula to compute equation of the axis of symmetry $x=-\frac{b}{2\times a}$ .

Axis of symmetry is

  $x=3$

Calculation :

Equating the two equations above.

  $-\frac{b}{2\times a}=3$

Simplifying the expression.

  $b=-6a$

In the above expression $a$ has to be negative as the parabola is downward facing

Putting $a=-1$ .

  $b=-6(-1)$

  $a{x}^2+bx+c$

In quadratic form,

  $y=-x^2+6x+16$

Here, c is the y-intercept.

Putting $x=8$ in the above expression,

  $y=-(8^2)+6\times 8+16$

Simplifying.

  $y=0$

This corresponds to the x-intercept $(8,0)$