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
1
1
9
2
17
3
25

Question content area bottom
Part 1
Represent the relationship using words. Choose the correct answer below.

Answer 1

To represent the relationship between the values of \(x\) and \(y\) in words, let's first summarize the data provided in the table:

• When \(x = 0\), \(y = 1\)
• When \(x = 1\), \(y = 9\)
• When \(x = 2\), \(y = 17\)
• When \(x = 3\), \(y = 25\)

Part 1: Word Representation

The relationship can be described as follows:

"As the value of \(x\) increases, the value of \(y\) changes according to a specific pattern. Specifically, for every increase of 1 in \(x\), \(y\) increases by 8. This suggests that \(y\) is dependent on \(x\) in a consistent manner."

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Next Steps

For parts 2 and 3 regarding the representation as an equation and a graph, we can first analyze the changes in \(y\) relative to changes in \(x\):

Change in \(y\):

1. From \(1\) to \(9\) (when \(x\) goes from \(0\) to \(1\)):
   • Change in \(y = 9 - 1 = 8\)
2. From \(9\) to \(17\) (when \(x\) goes from \(1\) to \(2\)):
   • Change in \(y = 17 - 9 = 8\)
3. From \(17\) to \(25\) (when \(x\) goes from \(2\) to \(3\)):
   • Change in \(y = 25 - 17 = 8\)

Consistent Change

The change in \(y\) is consistent, which indicates a linear relationship.

Part 2: Equation Representation

Since the change in \(y\) is consistently \(8\) for each increase of \(1\) in \(x\), this suggests that the equation of the relationship can be written in the slope-intercept form \(y = mx + b\).

Given the changes:

• The slope (\(m\)) is \(8\).
• Utilizing the point \( (0, 1) \) to find \(b\):

When \(x = 0\), \(y = 1\): \[ 1 = 8(0) + b \implies b = 1 \]

Thus, the equation is: \[ y = 8x + 1 \]

Part 3: Graph Representation

To graph this representation:

• Plot the points from the data: (0, 1), (1, 9), (2, 17), and (3, 25).
• Draw a straight line through these points, as they indicate a linear relationship.

Part 4: Determine if the relationship is a linear function

Since the changes in \(y\) relative to \(x\) are consistent, and the graph is a straight line, we can conclude that the relationship is indeed a linear function.

Thus, summarizing all parts, we have:

• Words: As \(x\) increases, \(y\) increases consistently by 8 for each increase of 1 in \(x\).
• Equation: \(y = 8x + 1\)
• Graph: A straight line through the plotted points \((0, 1)\), \((1, 9)\), \((2, 17)\), \((3, 25)\).

The relationship is a linear function.