Chapter 9.1, Problem 22PPS

To determine

To Graph and Find:

Use a table of values to graph $y=x^2+4x+6$ and state the domain and range.

Answer to Problem 22PPS

Domain is all real numbers.

Range = $\{y\in R:y\ge 2\}$ .

R = set of all real numbers

Explanation of Solution

Given Information:

Given equation is :

  $y=x^2+4x+6$

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.
• Parent Function:

  $y=x^2$

• Standard Form :

  $f(x)=a{x}^2+bx+c;a\ne 0$

• Type of Graph:

Parabola

• Axis of Symmetry:

  $x=-\frac{b}{2a}$

• y-intercept = 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.
• The range of the function will be all real numbers greater than or equal to the minimum value , or all real numbers less than or equal to the maximum value.

Calculation:

Make a table of values for :

  $y=x^2+4x+6$

x	$x^2+4x$	$y=x^2+4x+6$	(x,y)
-4	0	6	(-4,6)
-3	-3	3	(-3,3)
-2	-4	2	(-2,2)
-1	-3	3	(-1,3)
0	0	6	(0,6)

Graph the ordered pairs , and connect them to create a smooth curve .

The parabola extends to infinity.

  

For the domain, wee see x-coordinate values.Since, the graph takes all x values.

So, the domain is all real numbers.

For range, we see y-coordinate values. The range is all the real numbers greater than or equal to the minimum .So, the range is all real numbers greater than or equal to 2 = $\{y\in R:y\ge 2\}$ as the minimum value is 2.