QuestionIdentify the rule from the table:
1. $y=4x$
2. $y=x+3$
3. $y=x^2+3$
4. $y=4x^2$

Studdy Solution

STEP 1

Assumptions1. The table represents a mathematical rule that relates $x$ and $y$ . . The rule is consistent for all values of $x$ and $y$ in the table.
3. The rule is one of the four options given $y=x+3$ , $y=4x$ , $y=4x^{}$ , or $y=x^{}+3$ .

STEP 2

We will test each rule with the values given in the table. Let's start with the first rule, $y=x+$ .

STEP 3

Plug in the first pair of values from the table, $x=1$ and $y=$ , into the rule $y=x+3$ .
$y = x +3$ $=1 +3$

STEP 4

The first rule, $y=x+3$ , holds for the first pair of values. Now, let's test the second pair of values, $x=2$ and $y=7$ .
$y = x +3$ $7 =2 +3$

STEP 5

The rule $y=x+3$ does not hold for the second pair of values. Therefore, we can eliminate option A.

STEP 6

Now, let's test the second rule, $y=4x$ , with the first pair of values, $x=1$ and $y=4$ .
$y =4x$ $4 =4 \times1$

STEP 7

The second rule, $y=4x$ , holds for the first pair of values. Now, let's test the second pair of values, $x=2$ and $y=7$ .
$y =4x$ $7 =4 \times2$

STEP 8

The rule $y=4x$ does not hold for the second pair of values. Therefore, we can eliminate option B.

STEP 9

Now, let's test the third rule, $y=4x^{2}$ , with the first pair of values, $x=$ and $y=4$ .
$y =4x^{2}$ $4 =4 \times^{2}$

STEP 10

The third rule, $y=4x^{2}$ , does not hold for the first pair of values. Therefore, we can eliminate option C.

STEP 11

Now, let's test the last rule, $y=x^{}+3$ , with the first pair of values, $x=$ and $y=4$ .
$y = x^{}+3$ $4 =^{} +3$

STEP 12

The last rule, $y=x^{2}+$ , holds for the first pair of values. Now, let's test the second pair of values, $x=2$ and $y=7$ .
$y = x^{2}+$ $7 =2^{2} +$

STEP 13

The last rule, $y=x^{2}+3$ , holds for the second pair of values. Now, let's test the third pair of values, $x=3$ and $y=12$ .
$y = x^{2}+3$ $12 =3^{2} +3$

STEP 14

The last rule, $y=x^{2}+3$ , holds for the third pair of values. Now, let's test the fourth pair of values, $x=4$ and $y=19$ .
$y = x^{2}+3$ $19 =4^{2} +3$

STEP 15

The last rule, $y=x^{2}+3$ , holds for the fourth pair of values. Now, let's test the fifth pair of values, $x=5$ and $y=28$ .
$y = x^{2}+3$ $28 =5^{2} +3$

STEP 16

The last rule, $y=x^{2}+3$ , holds for the fifth pair of values. Therefore, the rule that the table represents is $y=x^{2}+3$ .
The rule represented by the table is $y=x^{2}+3$ .

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