Math Problem Statement
The table shows a function. Is the function linear or nonlinear?
Solution
To determine whether the function is linear or nonlinear, we need to check if the rate of change between the and values is constant.
Here’s the table from the image:
\hline x & y \\ \hline -10 & 10 \\ 8 \frac{2}{3} & -6 \\ 10 & -9 \\ \hline \end{array}$$ Let's calculate the differences in $$y$$ values for corresponding changes in $$x$$ values. A linear function will have a constant rate of change (constant slope). 1. From $$x = -10$$ to $$x = 8 \frac{2}{3}$$ (or $$\frac{26}{3}$$): - $$\Delta x = \frac{26}{3} - (-10) = \frac{26}{3} + \frac{30}{3} = \frac{56}{3}$$ - $$\Delta y = -6 - 10 = -16$$ 2. From $$x = 8 \frac{2}{3}$$ to $$x = 10$$: - $$\Delta x = 10 - \frac{26}{3} = \frac{30}{3} - \frac{26}{3} = \frac{4}{3}$$ - $$\Delta y = -9 - (-6) = -3$$ The rate of change for the first interval is $$\frac{\Delta y}{\Delta x} = \frac{-16}{\frac{56}{3}} = \frac{-16 \times 3}{56} = \frac{-48}{56} = -\frac{6}{7}$$. The rate of change for the second interval is $$\frac{\Delta y}{\Delta x} = \frac{-3}{\frac{4}{3}} = -3 \times \frac{3}{4} = -\frac{9}{4}$$. Since the rate of change is not constant, the function is **nonlinear**. Would you like further explanation or have any questions about this? Here are some related questions to explore further: 1. How do you determine the slope of a linear function? 2. What are the characteristics of a nonlinear function? 3. How can you graph a linear vs. nonlinear function? 4. What other methods can be used to check for linearity in functions? 5. How do you find the equation of a line given points? Tip: Always check the rate of change between points to determine if a function is linear.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Rate of Change
Slope
Linear and Nonlinear Functions
Formulas
Rate of change formula: Δy/Δx
Slope formula: (y2 - y1) / (x2 - x1)
Theorems
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Suitable Grade Level
Grades 8-10