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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.

x
y
0
negative 5
1
1
2
7
3
13

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

1 answer

To analyze the relationship between \( x \) and \( y \) based on the provided values, we first calculate the differences in \( y \) as \( x \) increases. Here's the data given:

\[ \begin{array}{|c|c|} \hline x & y \ \hline 0 & -5 \ 1 & 1 \ 2 & 7 \ 3 & 13 \ \hline \end{array} \]

Now, let's calculate the change in \( y \):

  • From \( x = 0 \) to \( x = 1 \): \( 1 - (-5) = 6 \)
  • From \( x = 1 \) to \( x = 2 \): \( 7 - 1 = 6 \)
  • From \( x = 2 \) to \( x = 3 \): \( 13 - 7 = 6 \)

The changes in \( y \) are consistent — each time \( x \) increases by 1, \( y \) increases by 6. This linearity suggests a linear function.

Part 1: Representing the Relationship Using Words

The appropriate representation from the choices given would be: A. The value of \( y \) is seven more than the quantity ten times the value of \( x \).

Part 2: Finding the Equation

Based on the increments, we can infer the relationship:

  1. The slope \( m \) (which is the change in \( y \) for a change in \( x \)) is \( 6 \).
  2. The base point when \( x = 0 \) gives \( y = -5 \).

With this, we can write the equation of the line in the slope-intercept form \( y = mx + b \):

\[ y = 6x - 5 \]

Part 3: Graphing the Function

To graph this linear equation, you can plot the points from the table:

  • (0, -5)
  • (1, 1)
  • (2, 7)
  • (3, 13)

Part 4: Determining if the Relationship is a Linear Function

Since the relationship is expressed with a constant slope (the differences in \( y \) values are constant when \( x \) increases), we can conclude that this is indeed a linear function.

Conclusion

  • In words: The value of \( y \) is seven more than the quantity ten times the value of \( x \).
  • The equation is \( y = 6x - 5 \).
  • The relationship is a linear function, confirmed by the constant slope.