Chapter 5, Problem 16MCQ

Solution

To describe the shape of the graph of given cubic function by determining the end behavior and number of turning points

The end behavior of $f\left(x\right)$ is down and up.

There is no turning point.

Given: $y=3x^3+4x^2+2x-1$

Concept Used:

End behavior of a polynomial function with leading term $a{x}^n$ is given by:-

• Up and up $[\text{When }a\text{ is positive, }n\text{ is even, }n\ne 0]$
• Down and down $[\text{When }a\text{ is negative, }n\text{ is even, }n\ne 0]$
• Down and up $[\text{When }a\text{ is positive, }n\text{ is odd}]$
• Up and down $[\text{When }a\text{ is negative, }n\text{ is odd}]$

Calculation:

Create table for values of x and y .

x	$3x^3+4x^2+2x-1$
-2	-13
-1	-2
0	-1
1	8
2	43

Plot points on the graph and join them.

  

The leading term of $f\left(x\right)$ is $3x^3$ , where $a=3\left(\text{positive}\right)$ and $n=3\left(\text{odd}\right).$

So, the end behavior of $f\left(x\right)$ is down and up.

The leading term of $f\left(x\right)$ is $x^7$ , where $a=1\left(\text{positive}\right)$ and $n=7\left(\text{odd}\right).$

So, the end behavior of $f\left(x\right)$ is down and up.

There is no turning point.