Math Problem Statement
Solution
Let's analyze the three functions based on the data provided and determine whether each is linear, quadratic, or exponential:
Function 1:
x & y \\ \hline 4 & 4 \\ 5 & -5 \\ 6 & 4 \\ 7 & 31 \\ 8 & 76 \\ \end{array}$$ - **Check if it's linear**: In a linear function, the difference between consecutive $$y$$-values (called the first differences) should be constant. - From $$x = 4$$ to $$x = 5$$, $$y$$ changes from 4 to -5 (difference of -9). - From $$x = 5$$ to $$x = 6$$, $$y$$ changes from -5 to 4 (difference of +9). - From $$x = 6$$ to $$x = 7$$, $$y$$ changes from 4 to 31 (difference of +27). - From $$x = 7$$ to $$x = 8$$, $$y$$ changes from 31 to 76 (difference of +45). Since the first differences are not constant, **it's not linear**. - **Check for quadratic**: A quadratic function has constant second differences. - First differences: $$-9, 9, 27, 45$$ - Second differences: $$9 - (-9) = 18$$, $$27 - 9 = 18$$, $$45 - 27 = 18$$ Since the second differences are constant (18), **Function 1 is quadratic**. ### Function 2: $$\begin{array}{c|c} x & y \\ \hline 0 & -3 \\ 1 & -10 \\ 2 & -21 \\ 3 & -36 \\ 4 & -55 \\ \end{array}$$ - **Check if it's linear**: - First differences: $$-10 - (-3) = -7$$, $$-21 - (-10) = -11$$, $$-36 - (-21) = -15$$, $$-55 - (-36) = -19$$ The first differences are not constant, so **it's not linear**. - **Check for quadratic**: - First differences: $$-7, -11, -15, -19$$ - Second differences: $$-11 - (-7) = -4$$, $$-15 - (-11) = -4$$, $$-19 - (-15) = -4$$ The second differences are constant (-4), so **Function 2 is quadratic**. ### Function 3: $$\begin{array}{c|c} x & y \\ \hline 5 & -5 \\ 6 & -9 \\ 7 & -13 \\ 8 & -17 \\ 9 & -21 \\ \end{array}$$ - **Check if it's linear**: - First differences: $$-9 - (-5) = -4$$, $$-13 - (-9) = -4$$, $$-17 - (-13) = -4$$, $$-21 - (-17) = -4$$ Since the first differences are constant, **Function 3 is linear**. ### Final Answers: - **Function 1**: Quadratic - **Function 2**: Quadratic - **Function 3**: Linear Would you like any further explanations or details on these functions? Here are five related questions to further explore this topic: 1. How do you derive a quadratic equation from given points? 2. What are the key characteristics of exponential functions compared to linear and quadratic ones? 3. Can a function be neither linear, quadratic, nor exponential? 4. How do the first and second differences help in identifying quadratic functions? 5. What happens to the rate of change in exponential functions? **Tip**: To identify exponential functions, check if the ratio of successive $$y$$-values is constant. This indicates a multiplicative pattern rather than an additive one.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Quadratic Functions
Exponential Functions
Formulas
Linear equation: y = mx + b
Quadratic equation: y = ax^2 + bx + c
Exponential function: y = a * b^x
Theorems
First differences indicate linearity
Second differences indicate a quadratic function
Suitable Grade Level
Grades 8-10
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