Chapter 9.1, Problem 77STP

To determine

To Find:

The range of given function.

Answer to Problem 77STP

Correct option is D.

Explanation of Solution

Given Information:

Given equation is :

  $y=-4x^2-\frac{1}{2}$

Options are:

A.{all integers less than or equal to $\frac{1}{2}$ }

B.{all non negative integers}

C.{all real numbers}

D.{all real numbers less than or equal to $-\frac{1}{2}$ }

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.
• 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.
• The range is all the real numbers greater than or equal to the minimum or all the real numbers less than or equal to the maximum.

Calculation:

• The given equation is $y=-4x^2-\frac{1}{2}$ , comparing it with the standard form of quadratic equation , we have a= -4 ,b= 0, c = $-\frac{1}{2}$.
• Here , a = -4 < 0 , so the graph opens downward and the function has maximum value at the vertex. So, the range will be all the real numbers less than or equal to the maximum value, which is at the vertex.
• Finding the vertex:

Since the axis of symmetry passes through the vertex , so the x-coordinate of the vertex is

  $x=-\frac{b}{2a}=-\frac{0}{2(-4)}=0$

Substitute the value of x = 0 in the given equation ,

  $\begin{array}{l}y=-4x^2-\frac{1}{2}\\ $ y=-4{(0)}^2-\frac{1}{2}\mathrm{...............}[x=0]$y=-\frac{1}{2}\end{array}$

Hence , the vertex is $(0,-\frac{1}{2})$

• So, the maximum value is $-\frac{1}{2}$
• Hence, the range is all the real numbers less than or equal to $-\frac{1}{2}$
• Correct option is D.