Question

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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
11
2
19
3
27

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

Answer 1

To represent the relationship in the table, we first need to analyze the data:

- When \( x = 0 \), \( y = 3 \)
- When \( x = 1 \), \( y = 11 \)
- When \( x = 2 \), \( y = 19 \)
- When \( x = 3 \), \( y = 27 \)

Let's observe the differences in \( y \) values as \( x \) increases:

- From \( x = 0 \) to \( x = 1 \): \( 11 - 3 = 8 \)
- From \( x = 1 \) to \( x = 2 \): \( 19 - 11 = 8 \)
- From \( x = 2 \) to \( x = 3 \): \( 27 - 19 = 8 \)

The difference in \( y \) is constant, which suggests a linear relationship. To find the equation of the line, we can use the form \( y = mx + b \).

**Calculating the slope (m):**
Since the increase in \( y \) is consistently 8 for each increase of 1 in \( x \), we find:
- \( m = 8 \)

Now we will substitute one of the points (for example, \( (0, 3) \)) into the equation to find \( b \):
- \( 3 = 8(0) + b \) -> \( b = 3 \)

**The equation is:**
\[
y = 8x + 3
\]

**Now, let's express the relationship using words:**
The value of \( y \) is eight times the value of \( x \) plus three.

**Choosing the correct answer from the options:**
None of the provided word options accurately match the relationship derived. The closest answer would likely be related to the coefficient of \( x \) which is 8, however, since none of the statements match, make note of that.

**Graphing:**
To graph the equation \( y = 8x + 3 \), you can plot the points from the table and connect them with a straight line.

**Determining if the relationship is a linear function:**
Yes, this relationship is indeed a linear function since the change in \( y \) corresponding to a change in \( x \) is constant.

In summary:
- The relationship in words: "The value of \( y \) is eight times the value of \( x \) plus three."
- The equation: \( y = 8x + 3 \)
- The relationship is a linear function.