Leslie analyzed the graph to determine if the function it represents is linear or non-linear. First, she found three points on the graph to be $(-1,-4),(0,-3)$, and $(2,5)$. Next, she determined the rate of change between the points $(-1,-4)$ and $(0,-3)$ to be $\frac{-3-(-4)}{0-(-1)}=\frac{1}{1}=1$ and the rate of change between the points $(0,-3)$ and $(2,5)$ to be $\frac{5-(-3)}{2-0}=\frac{8}{2}=4$. Finally, she concluded that since the rate of change is not constant, the function must be linear.
Why is Leslie wrong?
A. The points $(-1,-4),(0,-3)$, and $(2,5)$ are not all on the graph.
B. The expressions $\frac{-3-(-4)}{0-(-1)}$ and $\frac{5-(-3)}{2-0}$ both equal 1.
C. She miscalculated the rates of change.
D. Her conclusion is wrong. If the rate of change is not constant, then the function cannot be linear.
Answer (1)
Leslie incorrectly concluded that a function is linear when the rate of change between points is not constant. A linear function requires a constant rate of change. Since the calculated rates of change are different, the function is non-linear. Therefore, the correct answer is: Her conclusion is wrong. If the rate of change is not constant, then the function cannot be linear.
Explanation
Understanding the Problem Leslie wants to determine if a function is linear or non-linear based on three points: (-1, -4), (0, -3), and (2, 5). She calculates the rate of change between these points and concludes that the function is linear because the rate of change is not constant. We need to identify why her conclusion is wrong.
Linear vs. Non-linear Functions A linear function has a constant rate of change. This means that the slope between any two points on the line is the same. If the rate of change is not constant, the function is non-linear.
Calculating Rates of Change Leslie correctly calculated the rate of change between the points (-1, -4) and (0, -3) as: 0 − ( − 1 ) − 3 − ( − 4 ) = 0 + 1 − 3 + 4 = 1 1 = 1 She also correctly calculated the rate of change between the points (0, -3) and (2, 5) as: 2 − 0 5 − ( − 3 ) = 2 5 + 3 = 2 8 = 4
Identifying the Error Since the rate of change between the points is not constant (1 ≠ 4), the function is non-linear. Leslie's mistake is concluding that a non-constant rate of change implies the function is linear.
Final Answer Therefore, Leslie is wrong because her conclusion is incorrect. If the rate of change is not constant, the function cannot be linear.
Examples
Imagine you are tracking the distance a car travels over time. If the car moves at a constant speed, the relationship between time and distance is linear. However, if the car speeds up or slows down, the relationship becomes non-linear. Determining whether the relationship is linear or non-linear helps you understand the car's motion.
