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, minus, 4, comma, 2, right parenthesis, comma, left parenthesis, minus, 1, comma, 4, right parenthesis, comma, left parenthesis, 2, comma, 7, right parenthesis, comma, left parenthesis, 5, comma, 9, right parenthesis, right curly bracket
A=
{(−4,2),(−1,4),(2,7),(5,9)}
B, equals, left curly bracket, left parenthesis, 0, comma, 9, right parenthesis, comma, left parenthesis, 2, comma, 6, right parenthesis, comma, left parenthesis, 5, comma, 3, right parenthesis, comma, left parenthesis, 8, comma, 0, right parenthesis, right curly bracket
B=
{(0,9),(2,6),(5,3),(8,0)}
C, equals, left curly bracket, left parenthesis, minus, 2, comma, minus, 2, right parenthesis, comma, left parenthesis, 0, comma, 0, right parenthesis, comma, left parenthesis, 2, comma, 2, right parenthesis, comma, left parenthesis, 5, comma, 5, right parenthesis, right curly bracket
C=
{(−2,−2),(0,0),(2,2),(5,5)}
D, equals, left curly bracket, left parenthesis, minus, 4, comma, minus, 6, right parenthesis, comma, left parenthesis, 0, comma, minus, 5, right parenthesis, comma, left parenthesis, 4, comma, minus, 3, right parenthesis, comma, left parenthesis, 8, comma, minus, 1, right parenthesis, right curly bracket
D=
{(−4,−6),(0,−5),(4,−3),(8,−1)}

Answer 1

To determine which set of ordered pairs represents a linear function, we need to check if the points can be connected by a straight line. This can be done by seeing whether there is a constant rate of change (slope) between the pairs.

1. **Set A: { (−4, 2), (−1, 4), (2, 7), (5, 9) }**
- Slope between (−4, 2) and (−1, 4):
\[ m = \frac{4 - 2}{-1 - (-4)} = \frac{2}{3} \]
- Slope between (−1, 4) and (2, 7):
\[ m = \frac{7 - 4}{2 - (-1)} = \frac{3}{3} = 1 \]
- The slopes are not constant; therefore, **Set A does not represent a linear function**.

2. **Set B: { (0, 9), (2, 6), (5, 3), (8, 0) }**
- Slope between (0, 9) and (2, 6):
\[ m = \frac{6 - 9}{2 - 0} = \frac{-3}{2} \]
- Slope between (2, 6) and (5, 3):
\[ m = \frac{3 - 6}{5 - 2} = \frac{-3}{3} = -1 \]
- The slopes are not constant; therefore, **Set B does not represent a linear function**.

3. **Set C: { (−2, −2), (0, 0), (2, 2), (5, 5) }**
- Slope between (−2, -2) and (0, 0):
\[ m = \frac{0 - (-2)}{0 - (-2)} = \frac{2}{2} = 1 \]
- Slope between (0, 0) and (2, 2):
\[ m = \frac{2 - 0}{2 - 0} = \frac{2}{2} = 1 \]
- Slope between (2, 2) and (5, 5):
\[ m = \frac{5 - 2}{5 - 2} = \frac{3}{3} = 1 \]
- All the slopes are constant; therefore, **Set C represents a linear function**.

4. **Set D: { (−4, −6), (0, −5), (4, −3), (8, −1) }**
- Slope between (−4, -6) and (0, -5):
\[ m = \frac{-5 - (-6)}{0 - (-4)} = \frac{1}{4} \]
- Slope between (0, -5) and (4, -3):
\[ m = \frac{-3 - (-5)}{4 - 0} = \frac{2}{4} = \frac{1}{2} \]
- The slopes are not constant; therefore, **Set D does not represent a linear function**.

Based on this analysis, we find that **only Set C represents a linear function**.