Chapter 2.FR3, Problem 1E

Locating the Vertex of a Parabola In Exercises 1 through 4, find the vertex of the parabola that is the graph of the given quadratic function of $x.$ (Give the horizontal and vertical coordinates.) Determine whether the vertex represents a maximum or minimum for the function.

$x^2+6x-4$

To determine

The vertex of the parabola and to graph the given quadratic function $x^2+6x-4$ and also to determine whether the vertex represents a maximum or a minimum for the function.

Answer to Problem 1E

Solution:

Vertex of the parabola occurs at the point $(-3,-13)$ and the vertex represents a minimum for the function $x^2+6x-4$.

Explanation of Solution

Approach:

The graph of quadratic functions is a parabola. A quadratic expression has the form $a{x}^2+bx+c.$ Here $a$ is called the leading coefficient. If $a$ is positive, the graph is a parabola that opens upward and has a minimum value but no maximum value.

Ex: $x^2+2x+3$

It is in the form $a{x}^2+bx+c.$

$a=1,b=2,c=3$

We refer this vertex as maximum or minimum point of a parabola. The vertex of the parabola is given as $x=-\frac{b}{2a}.$

$x=-\frac{2}{2\times 1}=-\frac{2}{2}=-1$

Vertex is at $x=-1$

As the leading coefficient $a=1,$ is positive, it has a minimum value for the function $x^2+2x+3.$

If $a$ is negative, the graph is a parabola that opens downwards and has a maximum value but no minimum value.

Ex: ${-x}^2+2x+3$

It is in the form $a{x}^2+bx+c.$

$a=-1,b=2,c=3$

We refer this vertex as maximum or minimum point of a parabola. The vertex of the parabola is given as $x=-\frac{b}{2a}.$

$x=-\frac{2}{2\times -1}=-\frac{2}{-2}=1$

Vertex is at $x=1$

As the leading coefficient $a=-1,$ is negative, it has a maximum value for the function ${-x}^2+2x+3$

Calculation:

Given function is $x^2+6x-4$

It is in the form of quadratic function $a{x}^2+bx+c$.

$a=1,b=6,c=-4$

Vertex is at $x= -\frac{b}{2a}$

$=-\frac{6}{2\times 1}$

$=-\frac{6}{2}$

$=-3.$

Since the leading coefficient $a=1,$ is positive, it has a minimum value.

Final statement:

The vertex of the parabola occurs at a point $(-3,-13)$ and it represents a minimum for the function $x^2+6x-4.$