Math Problem Statement
Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
Solution
Let's analyze the function from the given table to determine whether it is linear or nonlinear.
Given points:
x & y = f(x) \\ \hline 0 & -25 \\ 1 & -1 \\ 2 & 7 \\ 3 & 3 \\ 4 & 16 \\ \end{array}$$ ### To check for linearity: A linear function has a constant slope between all points. The slope between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is calculated as: $$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$$ Let's calculate the slopes between consecutive points: 1. From $$(0, -25)$$ to $$(1, -1)$$: $$\text{slope}_1 = \frac{-1 - (-25)}{1 - 0} = \frac{24}{1} = 24$$ 2. From $$(1, -1)$$ to $$(2, 7)$$: $$\text{slope}_2 = \frac{7 - (-1)}{2 - 1} = \frac{8}{1} = 8$$ 3. From $$(2, 7)$$ to $$(3, 3)$$: $$\text{slope}_3 = \frac{3 - 7}{3 - 2} = \frac{-4}{1} = -4$$ 4. From $$(3, 3)$$ to $$(4, 16)$$: $$\text{slope}_4 = \frac{16 - 3}{4 - 3} = \frac{13}{1} = 13$$ Since the slopes between consecutive points are not the same (24, 8, -4, and 13), the function is **nonlinear**. ### Conclusion: The given function is nonlinear. Do you have any questions or want further clarification? Here are some related questions: 1. What is the definition of a linear function? 2. How can you graphically identify if a function is linear or nonlinear? 3. What are other types of functions besides linear and nonlinear? 4. Can a function have a varying slope and still be classified as linear? 5. How do you calculate the slope of a curve at a specific point? **Tip**: Always check for a constant rate of change (slope) between points to determine if a function is linear.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear and Nonlinear Functions
Slopes
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Theorems
Linearity in algebra: A function is linear if its rate of change (slope) is constant.
Suitable Grade Level
Grades 8-10
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