Chapter 6.4, Problem 42PPS

(a)

To determine

To graph: For the given function.

Explanation of Solution

Given information:

The given equation is $y=x^4-8x^2+16$

Graph:

Using MAPLE to graph the given function using the ranges as shown

  

Figure (1)

Interpretation: Graph for the function

  $f\left(x\right)=x^4-8x^2+16$ is shown in figure (1).

The number of zeroes is the number of intersection made on x- axis. From the above graph the number of intersection made is two

Therefore, the number of zeroes is 2 And the zero is $x=-2,0,2$ .

  

Figure 2

Therefore, the x-intercept is $-2$ and 2 , and the y intercept is 16 , and the turning points are made on $x=-2,0,2$ .

(b)

To determine

To find: The axis of symmetry.

Answer to Problem 42PPS

The graph of the line symmetry is $x=0$

Explanation of Solution

Given information:

The given equation is $f\left(x\right)=x^4-8x^2+16$

Calculation:

Line symmetry is the line which divides the given graph into equal halves

Therefore, clearly from the above graph, the line of symmetry is $x=0$ .

(c)

To determine

To find: The intervals for which it is increasing , decreasing ,or constant

Answer to Problem 42PPS

The graph is increasing in the interval $-\text{2}<\text{x}<0$ and $\text{x}>\text{2}$ and decreasing in and $\text{x}<-\text{2}$ and $0<\text{x}<\text{2}$

Explanation of Solution

Given information:

The given equation is $f\left(x\right)=x^4-8x^2+16$

Calculation:

The curve is increasing when the curve going upwards and is decreasing when he curve is going downwards.

Therefore, the graph is increasing in the interval $-\text{2}<\text{x}<0\text{ and x}>\text{2}$ decreasing in $\text{x}<-\text{2}$ and. $0<\text{x}<\text{2}$