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Question

Part 1
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
3
1
8
2
13
3
18

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Part 1
Represent the relationship using words. Choose the correct answer below.
A.
The value of y is ten less than the quantity nine times the value of x.
B.
The value of y is the square of the quantity one plus the value of x.
C.
The value of y is the cube of the quantity seven minus the value of x.
D.
The value of y is three more than the quantity five times the value of x.
Your answer is correct.E.
The value of y is ten less than the square of the quantity ten times the value of x.
F.
The value of y is six times the value of x.
Part 2
Represent the relationship using an equation.
yequals
  
enter your response here ​(Simplify your answer. Type an expression using x as the​ variable.)

Answer 1

To analyze the relationship between the values of \(x\) and \(y\) provided in the table, we first need to identify a pattern in the data.

Given values:

• When \(x = 0\), \(y = 3\)
• When \(x = 1\), \(y = 8\)
• When \(x = 2\), \(y = 13\)
• When \(x = 3\), \(y = 18\)

Part 1: Representing the relationship using words

From observing the values, we notice that \(y\) increases by 5 each time \(x\) increases by 1. Therefore, we can infer that \(y\) is a linear function of \(x\).

Let's analyze the options:

A. The value of y is ten less than the quantity nine times the value of x.
B. The value of y is the square of the quantity one plus the value of x.
C. The value of y is the cube of the quantity seven minus the value of x.
D. The value of y is three more than the quantity five times the value of x.
E. The value of y is ten less than the square of the quantity ten times the value of x.
F. The value of y is six times the value of x.

We can deduce that option D fits our observation best:

• The relationship can be expressed as: \( y = 5x + 3 \) (which we can verify by substituting the values).
• Thus, the correct answer is: D. The value of y is three more than the quantity five times the value of x.

Part 2: Representing the relationship using an equation

We can establish the relationship in equation form derived from our observation:

\[ y = 5x + 3 \]

This is a linear equation since it can be expressed in the form \( y = mx + b \), where \(m\) is the slope (which is 5) and \(b\) is the y-intercept (which is 3).

Conclusion

• The relationship is a linear function.
• The equation representing the relationship is:

\[ y = 5x + 3 \]