Chapter 9.1, Problem 4CYU

To determine

To Graph and Find:

Use a table of values to graph $y=3x^2-6x-5$ and state the domain and range.

Answer to Problem 4CYU

Domain is all real numbers.

Range = $\{y\in R:y\ge -8\}$ .

R = set of all real numbers

Explanation of Solution

Given Information:

Given equation is :

  $y=3x^2-6x-5$

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.

Calculation:

Make a table of values for :

  $y=3x^2-6x-5$

x	$3x^2-6x$	$y=3x^2-6x-5$	(x,y)
-1	9	4	(-1,4)
0	0	-5	(0,-5)
1	-3	-8	(1,-8)
2	0	-5	(2,-5)
3	9	4	(3,4)

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. Since, the graph takes all y values greater than or equal to y = -8.So, the range is all real numbers greater than or equal to -8 = $\{y\in R:y\ge -8\}$ as the minimum value is -8.