The graph of a function contains the points (−1, −4), (0, 1), (1, 4). Is the function linear? Explain.
Understand the Problem
The question is asking whether the given points form a linear function or not and requires an explanation for the answer.
Answer
The function is not linear because the graph containing the points is not a straight line.
Answer for screen readers
The function is not linear because the graph containing the points is not a straight line.
Steps to Solve
-
Identify the Points The given points are ((-1, -4)), ((0, 1)), ((1, 4)), and ((1, 4)).
-
Calculate Slopes Between Points To check if the points are linear, we calculate the slope between each pair of points.
-
Slope between ((-1, -4)) and ((0, 1)): $$ m_1 = \frac{1 - (-4)}{0 - (-1)} = \frac{5}{1} = 5 $$
-
Slope between ((0, 1)) and ((1, 4)): $$ m_2 = \frac{4 - 1}{1 - 0} = \frac{3}{1} = 3 $$
-
Slope between ((-1, -4)) and ((1, 4)): $$ m_3 = \frac{4 - (-4)}{1 - (-1)} = \frac{8}{2} = 4 $$
-
-
Compare the Slopes The slopes calculated are (5), (3), and (4). Since the slopes are not equal, this indicates that the points do not lie on the same straight line.
-
Conclusion on Linearity Since the slopes between the points are different, the points do not form a linear function.
The function is not linear because the graph containing the points is not a straight line.
More Information
In a linear function, any two points should yield the same slope when calculated. If the slopes differ, the points cannot lie on the same straight line, confirming that the relationship is not linear.
Tips
- Assuming that points are linear simply because they are plotted. Always calculate slopes to verify linearity.
- Not checking all pairs of points, which can lead to incomplete conclusions about whether the points are linear.
AI-generated content may contain errors. Please verify critical information
