Math Problem Statement
The table shows a function. Is the function linear or nonlinear?
Solution
The table you provided gives pairs of values for and as follows:
x & y \\ \hline 0 & 9 \\ 1 & 2 \\ 2 & 1 \\ \end{array}$$ To determine whether this function is linear or nonlinear, we check the rate of change of $$y$$ with respect to $$x$$. A function is linear if the rate of change (slope) is constant between all points. ### Step 1: Calculate the rate of change (slope) between consecutive points. - From $$(x_1 = 0, y_1 = 9)$$ to $$(x_2 = 1, y_2 = 2)$$: $$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 9}{1 - 0} = \frac{-7}{1} = -7$$ - From $$(x_2 = 1, y_2 = 2)$$ to $$(x_3 = 2, y_3 = 1)$$: $$\text{slope} = \frac{y_3 - y_2}{x_3 - x_2} = \frac{1 - 2}{2 - 1} = \frac{-1}{1} = -1$$ ### Step 2: Analyze the results. The slope between the points changes from $$-7$$ to $$-1$$, which means the rate of change is not constant. ### Conclusion: Since the rate of change is not constant, the function is **nonlinear**. Would you like more details or have any questions? Here are 5 follow-up questions to explore further: 1. What is a linear function, and how can you identify it from a table? 2. Can we determine if the function is quadratic based on the given data? 3. What other methods can be used to check if a function is nonlinear? 4. How does the graph of this function differ from a linear function graph? 5. Could this function have any special properties (e.g., exponential)? **Tip:** To quickly spot if a function is linear, check if the difference in $$y$$-values divided by the difference in $$x$$-values remains constant across all points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear vs Nonlinear Functions
Rate of Change
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Theorems
Criteria for linearity: constant rate of change
Suitable Grade Level
Grades 6-8