Chapter 9, Problem 64SGR

To determine

To determine the pattern in the table of values and subsequently write an equation that best represents the given data.

Answer to Problem 64SGR

The ordered pairs represent a quadratic function, whose equation can be modelled on $y=-x^2$ .

Explanation of Solution

Given information :

‘x’	0	1	2	3	4
‘y’	0	-1	-4	-9	-16

Formula used :

Find the first difference in the values of ‘y’ in the given data set. If the successive differences are same, the data set represents a straight line. If the differences are not same, use the second difference. If here the differences of the first set are constant, the data set represents a quadratic equation.

To compute the equation representing the lines, use the standard for of $y=a{x}^2$ . Replace any of the values in the given set and solve of ‘a’. This shall be the equation.

Calculation :

‘x’	0	1	2	3	4
‘y’	0	-1	-4	-9	-16

Given data set.

First difference

1

3

5

7

Subtract successive values of ‘y’ in order to find figures of the first difference.

For example, the first value of 1 is calculated by subtracting -1 from 0. That is, $0-\left(-1\right)=1$ .

Since the values of the first difference are not constant, compute second difference.

First difference

-2

-2

-2

Subtract successive values of the first difference to find the figures. For example, 6 is calculated by subtracting 3 from 1. That is, $1-\left(3\right)=-2$

Since the values of the second difference are constant, the data is modelled by a quadratic function. The equation is of the form $y=a{x}^2$ .

Value of ‘a’ can be found by replacing an ordered pair in the standard form $y=a{x}^2$ . Try using non-zero items in order to find the value.

Here, take $\left(3,-9\right)$ .

  $-9=a\times 3^2$

Putting an ordered pair in $y=a{x}^2$ .

  $-1=a$

Hence the equation is $y=-x^2$