Math Problem Statement
The table shows a function. Is the function linear or nonlinear? x y 2 1 1 3 4 2 6 6
Solution
To determine if a function is linear or nonlinear, we need to examine whether the change in corresponds consistently to the change in (a constant rate of change or slope).
Here is the given table:
\hline x & y \\ \hline 2 & 1 \\ ? & 1 \\ 3 & 4 \\ 4 & 2 \\ 6 & 6 \\ \hline \end{array}$$ ### Steps: 1. **Verify the completeness of the data**: The table has an unspecified value for $$x$$ when $$y = 1$$. If we cannot determine this missing value, we cannot fully conclude whether the function is linear or nonlinear. 2. **Check rate of change** (slope): - Calculate the slope between each pair of points using the formula: $$\text{slope} = \frac{\Delta y}{\Delta x}.$$ - If all slopes are equal, the function is linear. ### Observations: 1. Between $$(2, 1)$$ and $$(3, 4)$$: $$\text{slope} = \frac{4 - 1}{3 - 2} = \frac{3}{1} = 3.$$ 2. Between $$(3, 4)$$ and $$(4, 2)$$: $$\text{slope} = \frac{2 - 4}{4 - 3} = \frac{-2}{1} = -2.$$ 3. Between $$(4, 2)$$ and $$(6, 6)$$: $$\text{slope} = \frac{6 - 2}{6 - 4} = \frac{4}{2} = 2.$$ Since the slopes between points are not constant, the function is **nonlinear**. ### Missing Data: The unspecified point ($$x = ?$$, $$y = 1$$) does not affect the conclusion because the inconsistent slopes among the provided data points already establish nonlinearity. --- Would you like more details on how the calculations were performed? --- ### Related Questions: 1. What are the characteristics of a linear function? 2. How do you graph a nonlinear function based on a table? 3. Can the missing data point make the function linear in any case? 4. What real-world examples are modeled by nonlinear functions? 5. How can you identify a function's type from its equation? ### Tip: Always check for missing data in tables to ensure accurate interpretation of functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Functions
Linear Functions
Nonlinear Functions
Rate of Change
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Theorems
Definition of linearity (constant rate of change)
Suitable Grade Level
Grades 7-9
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