For the table to the​ right, represent the relationship using​ words, an​ equation, and a graph. Then determine whether the relationship is a linear function.

Question

x
y
0
3
1
11
2
19
3
27

Question content area bottom
Part 1
Represent the relationship using words. Choose the correct answer below.
A.
The value of y is eight times the value of x.
B.
The value of y is four less than the square of the quantity seven times the value of x.
C.
The value of y is the cube of the quantity six minus the value of x.
D.
The value of y is the square of the quantity eight plus the value of x.
E.
The value of y is eight less than the quantity four times the value of x.
F.
The value of y is three more than the quantity eight times the value of x.

Answer 1

To analyze the relationship between the values of \( x \) and \( y \) in the given table:

x | y
---------
0 | 3
1 | 11
2 | 19
3 | 27

Let's first observe the changes in \( y \) as \( x \) increases.

Calculating \( y \) values:

• From \( x = 0 \) to \( x = 1 \): \( y \) changes from 3 to 11 (a change of 8).
• From \( x = 1 \) to \( x = 2 \): \( y \) changes from 11 to 19 (a change of 8).
• From \( x = 2 \) to \( x = 3 \): \( y \) changes from 19 to 27 (a change of 8).

The change in \( y \) is consistent: for every increase of 1 in \( x \), \( y \) increases by 8. This indicates a linear relationship.

Part 1 - Representing the relationship in words:

From the analysis, we see that:

• For every increment of 1 in \( x \), \( y \) increases by 8. This suggests that \( y \) is proportional to \( x \) with some constant.

Thus, the correct choice that best describes this linear relationship is: E. The value of y is eight less than the quantity four times the value of x.

Part 2 - The equation:

From the pattern observed, we can derive the linear equation:

Given:

• The slope (change) \( m \) is 8.
• The \( y \)-intercept (when \( x = 0 \), \( y = 3 \)).

The equation of a linear function can be expressed as: \[ y = mx + b \] Substituting the values: \[ y = 8x + 3 \]

Part 3 - The graph:

To graph this equation, you can plot the points (0, 3), (1, 11), (2, 19), and (3, 27) and then draw a straight line through them.

Part 4 - Determine whether the relationship is a linear function.

Since we determined that the relationship between \( x \) and \( y \) produces a constant rate of change (the slope is constant), and we can express it as a linear equation, the relationship is indeed a linear function.

Thus, to summarize:

• It is a linear function.
• The verbal representation is E, the equation is \( y = 8x + 3 \), and it can be graphed as described.