Question

Which set of ordered pairs left parenthesis, x, comma, y, right parenthesis(x,y) could represent a linear function?
A, equals, left curly bracket, left parenthesis, 3, comma, minus, 4, right parenthesis, comma, left parenthesis, 5, comma, 2, right parenthesis, comma, left parenthesis, 6, comma, 4, right parenthesis, comma, left parenthesis, 7, comma, 6, right parenthesis, right curly bracket
A=
{(3,−4),(5,2),(6,4),(7,6)}
B, equals, left curly bracket, left parenthesis, minus, 2, comma, 4, right parenthesis, comma, left parenthesis, 0, comma, 3, right parenthesis, comma, left parenthesis, 3, comma, 2, right parenthesis, comma, left parenthesis, 5, comma, 1, right parenthesis, right curly bracket
B=
{(−2,4),(0,3),(3,2),(5,1)}
C, equals, left curly bracket, left parenthesis, minus, 2, comma, 7, right parenthesis, comma, left parenthesis, 0, comma, 5, right parenthesis, comma, left parenthesis, 1, comma, 3, right parenthesis, comma, left parenthesis, 2, comma, 1, right parenthesis, right curly bracket
C=
{(−2,7),(0,5),(1,3),(2,1)}
D, equals, left curly bracket, left parenthesis, 1, comma, 8, right parenthesis, comma, left parenthesis, 3, comma, 0, right parenthesis, comma, left parenthesis, 4, comma, minus, 4, right parenthesis, comma, left parenthesis, 5, comma, minus, 8, right parenthesis, right curly bracket
D=
{(1,8),(3,0),(4,−4),(5,−8)}
Answer
Multiple Choice Answers
AA
BB
CC
DD

Answer 1

To determine which set of ordered pairs represents a linear function, we need to check if the y-values change at a constant rate as the x-values change. A linear function is characterized by a constant slope.

Let's analyze each set of ordered pairs:

**A:**
{(3,−4),(5,2),(6,4),(7,6)}

Calculating the slope between each pair:

1. Between (3,−4) and (5,2):
\[
\text{slope} = \frac{2 - (-4)}{5 - 3} = \frac{2 + 4}{2} = \frac{6}{2} = 3
\]

2. Between (5,2) and (6,4):
\[
\text{slope} = \frac{4 - 2}{6 - 5} = \frac{2}{1} = 2
\]

3. Between (6,4) and (7,6):
\[
\text{slope} = \frac{6 - 4}{7 - 6} = \frac{2}{1} = 2
\]

The slopes are not constant; thus, **set A** does not represent a linear function.

---

**B:**
{(−2,4),(0,3),(3,2),(5,1)}

Calculating the slope:

1. Between (−2,4) and (0,3):
\[
\text{slope} = \frac{3 - 4}{0 - (-2)} = \frac{-1}{2}
\]

2. Between (0,3) and (3,2):
\[
\text{slope} = \frac{2 - 3}{3 - 0} = \frac{-1}{3}
\]

3. Between (3,2) and (5,1):
\[
\text{slope} = \frac{1 - 2}{5 - 3} = \frac{-1}{2}
\]

The slopes are not constant; thus, **set B** does not represent a linear function.

---

**C:**
{(−2,7),(0,5),(1,3),(2,1)}

Calculating the slope:

1. Between (−2,7) and (0,5):
\[
\text{slope} = \frac{5 - 7}{0 - (-2)} = \frac{-2}{2} = -1
\]

2. Between (0,5) and (1,3):
\[
\text{slope} = \frac{3 - 5}{1 - 0} = -2
\]

3. Between (1,3) and (2,1):
\[
\text{slope} = \frac{1 - 3}{2 - 1} = -2
\]

The slopes are not constant; thus, **set C** does not represent a linear function.

---

**D:**
{(1,8),(3,0),(4,−4),(5,−8)}

Calculating the slope:

1. Between (1,8) and (3,0):
\[
\text{slope} = \frac{0 - 8}{3 - 1} = \frac{-8}{2} = -4
\]

2. Between (3,0) and (4,−4):
\[
\text{slope} = \frac{-4 - 0}{4 - 3} = \frac{-4}{1} = -4
\]

3. Between (4,−4) and (5,−8):
\[
\text{slope} = \frac{-8 - (-4)}{5 - 4} = \frac{-4}{1} = -4
\]

The slopes are constant (-4); thus, **set D** represents a linear function.

Therefore, the correct answer is:

**D**