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
5
2
9
3
13

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

Answer 1

To represent the relationship in the table using words, we need to analyze the data first. The pairs of (x, y) values are:

• (0, 1)
• (1, 5)
• (2, 9)
• (3, 13)

Now, let's determine the pattern in the values of y with respect to x:

• When \( x = 0 \), \( y = 1 \)
• When \( x = 1 \), \( y = 5 \)
• When \( x = 2 \), \( y = 9 \)
• When \( x = 3 \), \( y = 13 \)

Looking at the change in y values for each increment in x:

• From \( x = 0 \) to \( x = 1 \), \( y\) increases by \( 4 \) (from 1 to 5).
• From \( x = 1 \) to \( x = 2 \), \( y\) increases by \( 4 \) (from 5 to 9).
• From \( x = 2 \) to \( x = 3 \), \( y\) increases by \( 4 \) (from 9 to 13).

The consistent increase indicates a linear relationship. The pattern suggests that \( y \) can be expressed as a linear equation involving \( x \).

Part 1: Represent the relationship using words.

The correct option that best fits our pattern is:

A. The value of y is one more than the quantity four times the value of x.

Part 2: Write an equation.

From our observation, the relationship can be expressed mathematically as:

\[ y = 4x + 1 \]

Part 3: Graph the relationship.

To graph the relationship, you would plot the points (0, 1), (1, 5), (2, 9), and (3, 13) on a coordinate grid. Then draw a straight line through these points, as they exhibit a linear relationship.

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

Since the difference in y values is constant as x increases, the relationship represented by the points indeed forms a straight line. Therefore, the relationship is a linear function.