Identify the two tables which represent quadratic relation ships

Question

Table 1:
x: 0, 1, 2, 3
y: 1, 2, 4, 8

Table 2:
x: 0, 1, 2, 3
y: -2, -4, -8, -16
Table 3:
x: 0, 1, 2, 3
y: -2, 0, 2, 4
Table:4
x: 0, 1, 2, 3
y: 3, 4, 5, 6
Table 5:
x: 0, 1, 2, 3
y: -4, -8, -10, -10
Table 6:
x: 0, 1, 2, 3
y: 4, -4, -4, 4

Answer 1

To identify which of the provided tables represent quadratic relationships, we need to analyze the relationship between the \(x\) and \(y\) values for each table and look for a consistent second difference in the \(y\) values when the \(x\) values increase by 1.

Let's calculate the first and second differences for each table:

Table 1

• \(x: 0, 1, 2, 3\)
• \(y: 1, 2, 4, 8\)

First differences:
\(2 - 1 = 1\)
\(4 - 2 = 2\)
\(8 - 4 = 4\)

\((1, 2, 4)\)

Second differences:
\(2 - 1 = 1\)
\(4 - 2 = 2\)

\((1, 2)\)

Table 2

• \(x: 0, 1, 2, 3\)
• \(y: -2, -4, -8, -16\)

First differences:
\(-4 - (-2) = -2\)
\(-8 - (-4) = -4\)
\(-16 - (-8) = -8\)

\((-2, -4, -8)\)

Second differences:
\(-4 - (-2) = -2\)
\(-8 - (-4) = -4\)

\((-2, -4)\)

Table 3

• \(x: 0, 1, 2, 3\)
• \(y: -2, 0, 2, 4\)

First differences:
\(0 - (-2) = 2\)
\(2 - 0 = 2\)
\(4 - 2 = 2\)

\((2, 2, 2)\)

Second differences:
\(2 - 2 = 0\)
\(2 - 2 = 0\)

\((0, 0)\)

Table 4

• \(x: 0, 1, 2, 3\)
• \(y: 3, 4, 5, 6\)

First differences:
\(4 - 3 = 1\)
\(5 - 4 = 1\)
\(6 - 5 = 1\)

\((1, 1, 1)\)

Second differences:
\(1 - 1 = 0\)
\(1 - 1 = 0\)

\((0, 0)\)

Table 5

• \(x: 0, 1, 2, 3\)
• \(y: -4, -8, -10, -10\)

First differences:
\(-8 - (-4) = -4\)
\(-10 - (-8) = -2\)
\(-10 - (-10) = 0\)

\((-4, -2, 0)\)

Second differences:
\(-2 - (-4) = 2\)
\(0 - (-2) = 2\)

\((2, 2)\)

Table 6

• \(x: 0, 1, 2, 3\)
• \(y: 4, -4, -4, 4\)

First differences:
\(-4 - 4 = -8\)
\(-4 - (-4) = 0\)
\(4 - (-4) = 8\)

\((-8, 0, 8)\)

Second differences:
\(0 - (-8) = 8\)
\(8 - 0 = 8\)

\((8, 8)\)

Summary of the Second Differences

• Table 1: Second differences are not constant.
• Table 2: Second differences are not constant.
• Table 3: Second differences are constant (0).
• Table 4: Second differences are constant (0).
• Table 5: Second differences are constant (2).
• Table 6: Second differences are constant (8).

Identified Quadratic Relationships

The quadratic relationships can be characterized by consistent second differences. Based on the analysis, only Table 5 shows constant second differences that suggest a quadratic relation.

Thus, the two tables that represent quadratic relationships are:

• Table 5
• Table 6

Tables 3 and 4 do not represent quadratic relationships because they have constant first differences, not second differences.