Math Problem Statement

Select the correct statement about the function represented by the table.

Solution

The table presented shows the relationship between xx and yy values, and the question asks you to determine whether the function is exponential or linear based on the pattern in the data.

Let’s break it down step by step:

Given dataThe table presented shows the relationship between xx and yy values, and the question asks you to determine whether the function is exponential or linear based on the pattern in the data.

Let’s break it down step by step:

Given

x & y \\ \hline 1 & 26 \\ 2 & 44 \\ 3 & 62 \\ 4 & 80 \\ 5 & 98 \\ \end{array}$$ ### Linear Function Test: In a linear function, the difference between consecutive $$y$$-values (called the first difference) should be constant. Let’s check: - $$44 - 26 = 18$$ - $$62 - 44 = 18$$ - $$80 - 62 = 18$$ - $$98 - 80 = 18$$ Since the differences are constant ($$18$$), the function increases by the same amount for each unit increase in $$x$$. This is a key characteristic of a **linear function**. ### Exponential Function Test: In an exponential function, the ratio between consecutive $$y$$-values (i.e., the factor of growth) should be constant. Let’s check: - $$\frac{44}{26} \approx 1.692$$ - $$\frac{62}{44} \approx 1.409$$ - $$\frac{80}{62} \approx 1.290$$ - $$\frac{98}{80} \approx 1.225$$ Since these ratios are not constant, the function is **not exponential**. ### Conclusion: The correct answer is **B**: "It is a linear function because the $$y$$-values increase by an equal difference over equal intervals of $$x$$-values." Would you like any more details or clarification? Here are five related questions to consider: 1. What distinguishes an exponential function from a linear function? 2. How do you determine the growth rate of an exponential function? 3. Can a function be both linear and exponential? Why or why not? 4. How would the function change if the differences between \(y

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Exponential Functions
Difference Test
Ratio Test

Formulas

First difference between consecutive y-values in a linear function: Δy = y2 - y1
Ratio between consecutive y-values in an exponential function: r = y2 / y1

Theorems

Linear Function: The difference between consecutive y-values is constant for a linear function.
Exponential Function: The ratio between consecutive y-values is constant for an exponential function.

Suitable Grade Level

Grades 8-10

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