Chapter 9.7, Problem 21PPE

Solution

(a)

To calculate: The common second difference of the functions.

The required second difference is $-x^2-5x-3$ .

Given Information:

The given functions are $f\left(x\right)=x^2-3,f\left(x\right)=3x^2\text{and}f\left(x\right)=4x^2-5x$ .

Calculation:

Consider the given functions $f\left(x\right)=x^2-3,f\left(x\right)=3x^2\text{and}f\left(x\right)=4x^2-5x$ .

Now, subtract the function $f\left(x\right)=x^2-3$ from $f\left(x\right)=3x^2$

  $3x^2-x^2-3=2x^2+3$  ...... (1)

Similarly, subtract the function $f\left(x\right)=3x^2$ from $f\left(x\right)=4x^2-5x$ .

  $4x^2-5x-3x^2=x^2-5x$  ...... (2)

Now find the second difference, subtract (1) from (2).

  $x^2-5x-2x^2-3=-x^2-5x-3$

Hence, the required second difference is $-x^2-5x-3$ .

(b)

To find: The relation between the common second difference and the coefficient of the term $x^2$ .

The value of the coefficient indicates the scaling of the graph and the quadrant of the graph.

Given Information:

The given common second difference is $-x^2-5x-3$ .

Calculation:

Consider the given common second difference is $-x^2-5x-3$ .

Here the coefficient of the term $x^2$ is $-1$ .

This coefficient indicates the scaling of the graph and the quadrant of the graph.

Hence, the value of the coefficient indicates the scaling of the graph and the quadrant of the graph.