Math Problem Statement
Which of the following statements is consistent with the table of values? (Given the table with x = [2, 3, 4, 5] and f(x) = [4, 9, 18, 31])
Solution
The question is asking about the table of values for a function and which expression for the average rate of change fits the table. Here's how we can break it down:
Given the table:
\hline x & 2 & 3 & 4 & 5 \\ f(x) & 4 & 9 & 18 & 31 \\ \hline \end{array}$$ We need to find the average rate of change (AROC) between different points and see if the function appears linear or quadratic. Here's the approach: ### Step 1: Calculate the average rate of change between each pair of points: - Between $$x = 2$$ and $$x = 3$$: $$\frac{f(3) - f(2)}{3 - 2} = \frac{9 - 4}{1} = 5$$ - Between $$x = 3$$ and $$x = 4$$: $$\frac{f(4) - f(3)}{4 - 3} = \frac{18 - 9}{1} = 9$$ - Between $$x = 4$$ and $$x = 5$$: $$\frac{f(5) - f(4)}{5 - 4} = \frac{31 - 18}{1} = 13$$ ### Step 2: Analyze the rate of change The average rate of change is increasing as $$x$$ increases, which indicates that this function is not linear (for a linear function, the rate of change would be constant). Therefore, the function is quadratic. ### Step 3: Match the correct answer The correct expression from the options is: **"The expression for the average rate of change is 5a - 6, and $$f$$ is quadratic."** This corresponds to the answer you selected, which is correct based on the table and the increasing rate of change. Would you like a more detailed explanation or have any other questions? Here are 5 related questions to expand on this topic: 1. How is the average rate of change related to the derivative in calculus? 2. What is the general form of a quadratic function? 3. How can you use differences in values to identify a quadratic function from a table? 4. What is the formula for finding the average rate of change between two points on a curve? 5. How do you find the exact equation of a quadratic function from a set of points? **Tip:** When analyzing tables of values, a constant second difference (difference of differences) is a good indication that the function is quadratic.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Average Rate of Change
Quadratic Functions
Linear Functions
Formulas
Average Rate of Change: AROC = (f(b) - f(a)) / (b - a)
Theorems
Quadratic Function Theorem
Linear Function Properties
Suitable Grade Level
Grade 9-10
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