Exercise 4 Page 96

We want to find the pattern between the values and fill in the missing entries in the table below.

IN (x)	OUT (y)
2	5
9	$ $
- 2	- 5
$ $	- 7.5
0	$ $
$ $	27.5
- 4	- 10

Let's focus again on the two inputs 2 and - 2. We are going to evaluate the difference between these inputs and the difference between their corresponding outputs. Difference of Inputs:& 2 - (- 2)=4 Difference of Outputs:& 5 - (- 5)=10We see that the difference between the outputs is 2.5 times larger than the difference between the inputs. 10/4=2.5 Let's check if the inputs multiplied by 2.5 give the outputs from the table.

IN (x)	2.5x	OUT (y)
2	2.5(2)=5	5
- 2	2.5(- 2)=- 5	- 5
- 4	2.5(- 4)=- 10	- 10

Since the results match, multiplying the input by 2.5 gives us the output. 2.5* INPUT= OUTPUT This is our pattern. Let's now fill in the missing gaps, starting with the outputs.

IN (x)	OUT (y)
2	5
9	2.5( 9)= 22.5
- 2	- 5
$ $	- 7.5
0	2.5( 0)= 0
$ $	27.5
- 4	- 10

Now we need to reverse the pattern to find the missing input values. Earlier we multiplied the input value by 2.5, so now we need to divide the output by 2.5. This will give us the input values. 2.5* INPUT&= OUTPUT &⇓ INPUT &= OUTPUT/2.5 We are ready to complete our table.

IN (x)	OUT (y)
2	5
9	22.5
- 2	- 5
- 7.5/2.5 = - 3	- 7.5
0	0
27.5/2.5= 11	27.5
- 4	- 10

The table was created by multiplying the input by 2.5.

Our goal is to find the pattern between the values and fill in the missing entries in the table below.

IN (x)	OUT (y)
1	$ $
$ $	4
4	3
- 20	- 9
- 10	$ $
0	1
$ $	- 2.5

Once again, let's consider two inputs, 4 and - 20. We need to evaluate the differences between these inputs and between their corresponding outputs. Difference of Inputs:& 4 - (- 20)=24 Difference of Outputs:& 3 - (- 9)=12We see that the difference between the outputs is half the difference between the inputs. 12/24=0.5 Let's check whether the inputs multiplied by 0.5 give the outputs from the table.

IN (x)	0.5x	OUT (y)
4	0.5(4)=2	3
- 20	0.5(- 20)=- 10	- 9
0	0.5(0)=0	1

We see that the input multiplied by 0.5 is always 1 less than the desired output, so let's also add 1 to our rule and see if the results match.

IN (x)	0.5x+1	OUT (y)
4	0.5(4)+1=3	3
- 20	2.5(- 20)+1=- 9	- 9
0	0.5(0)+1=10	1

Since the results match, multiplying the input by 0.5 and adding 1 gives us the output! 0.5* INPUT+1= OUTPUT This is our pattern. Let's now fill in the missing gaps, starting with the outputs.

IN (x)	OUT (y)
1	0.5( 1)+1= 1.5
$ $	4
4	3
- 20	- 9
- 10	0.5( - 10)+1= - 4
0	1
$ $	- 2.5

Now, in order to find the missing inputs, we will reverse the pattern. 0.5* INPUT+1&= OUTPUT &⇓ INPUT &= OUTPUT-1/0.5 Let's complete our table.

IN (x)	OUT (y)
1	1.5
4-1/0.5= 6	4
4	3
- 20	- 9
- 10	- 4
0	1
- 2.5-1/0.5= - 7	- 2.5

To fill in the table, we multiplied the input by 0.5 and added 1.