Math Problem Statement
The table shows a function. Is the function linear or nonlinear?
Solution
To determine if the function in the table is linear or nonlinear, we need to check if the rate of change between the pairs of and values is constant.
Given table:
x & y \\ \hline -6 & -10 \\ -1 & 15 \\ 4 & 20 \\ \end{array}$$ ### Step 1: Calculate the rate of change (slope) between points. The formula for slope $$m$$ between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ #### Between (-6, -10) and (-1, 15): $$m_1 = \frac{15 - (-10)}{-1 - (-6)} = \frac{15 + 10}{-1 + 6} = \frac{25}{5} = 5$$ #### Between (-1, 15) and (4, 20): $$m_2 = \frac{20 - 15}{4 - (-1)} = \frac{5}{4 + 1} = \frac{5}{5} = 1$$ ### Step 2: Analyze the results. The slope between the first two points is 5, while the slope between the next two points is 1. Since the slopes are not consistent, the rate of change is not constant. ### Conclusion: The function is **nonlinear**. Do you have any further questions or would you like more details? --- Here are 5 related questions to expand the topic: 1. How can you identify if a function is linear based on its equation? 2. What is the difference between linear and nonlinear functions? 3. How do you calculate the slope for more complex functions? 4. What is the significance of the slope in a linear function? 5. Can a nonlinear function have a constant slope at certain intervals? **Tip:** To recognize a linear function from a table, the change in $$y$$ values must be proportional to the change in $$x$$ values across all intervals.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear and Nonlinear Functions
Rate of Change
Formulas
Slope formula: m = (y2 - y1) / (x2 - x1)
Theorems
A function is linear if the rate of change is constant.
Suitable Grade Level
Grades 7-9
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