Question

1. which equation corresponds to the value pairs in the table below?

x	y
1	4
5	28
7	52
9	84
10	103
100	10,003

options:

• $y = x^2$
• $y = x^2 + 3$
• $y = 10x + 3$
• $y = x + 3$

Explanation:

Step1: Test \( x = 0.5 \)

Substitute \( x = 0.5 \) into \( y = x^3 + 3 \): \( y = (0.5)^3 + 3 = 0.125 + 3 = 3.125 \)? Wait, no, wait the table has \( y = 3.25 \). Wait, maybe I miscalculated. Wait \( (0.5)^3 = 0.125 \), \( 0.125 + 3 = 3.125 \), but table has 3.25. Wait maybe the option is \( y = x^3 + 3 \)? Wait no, let's check \( x = 1 \). For \( x = 1 \), \( y = 1^3 + 3 = 4 \), which matches the table (1,4). Now \( x = 5 \): \( 5^3 + 3 = 125 + 3 = 128 \)? No, table has 28. Wait, wait maybe I misread the options. Wait the options are \( y = x^2 \), \( y = x^3 + 3 \), \( y = 10x + 3 \), \( y = x + 3 \). Wait for \( x = 1 \), \( y = 4 \). Let's check \( y = x^3 + 3 \): \( 1^3 + 3 = 4 \), correct. \( x = 5 \): \( 5^3 + 3 = 128 \), but table has 28. Wait that's not matching. Wait maybe the option is \( y = x^2 + 3 \)? Wait the second option is \( y = x^2 + 3 \)? Wait the user's image: the second option is \( y = x^2 + 3 \)? Wait let's recheck. For \( x = 0.5 \), \( x^2 = 0.25 \), \( 0.25 + 3 = 3.25 \), which matches the table (0.5, 3.25). \( x = 1 \): \( 1 + 3 = 4 \), matches. \( x = 5 \): \( 25 + 3 = 28 \), matches! \( x = 7 \): \( 49 + 3 = 52 \), matches. \( x = 9 \): \( 81 + 3 = 84 \), matches. \( x = 10 \): \( 100 + 3 = 103 \), matches. \( x = 100 \): \( 10000 + 3 = 10003 \), matches. Oh! So the correct equation is \( y = x^2 + 3 \). Wait the second option is \( y = x^2 + 3 \) (maybe the user wrote \( x^2 \) as \( x^2 \)). So the answer is the second option, \( y = x^2 + 3 \).

Step2: Verify with \( x = 0.5 \)

\( x = 0.5 \), \( y = (0.5)^2 + 3 = 0.25 + 3 = 3.25 \), matches the table.

Step3: Verify with \( x = 5 \)

\( x = 5 \), \( y = 5^2 + 3 = 25 + 3 = 28 \), matches the table.

Step4: Verify with \( x = 100 \)

\( x = 100 \), \( y = 100^2 + 3 = 10000 + 3 = 10003 \), matches the table.

So the correct equation is \( y = x^2 + 3 \), which is the second option.

Answer:

\( y = x^3 + 3 \)