Math Problem Statement
Which table of values represents a linear function? A xx yy 33 11 55 33 77 66 99 88 B xx yy 22 44 33 22 55 00 77 minus, 2−2 C xx yy 00 00 22 33 44 66 66 88 D xx yy 00 44 22 33 44 22 66 11
Solution
A linear function is one where the difference in the y-values (outputs) for each unit change in the x-values (inputs) is constant. Let's analyze each table and see which one represents a linear relationship.
Table A:
\hline x & y \\ \hline 3 & 11 \\ 5 & 33 \\ 7 & 66 \\ 9 & 88 \\ \hline \end{array}$$ - The difference between consecutive $$x$$-values is constant: $$5 - 3 = 2$$, $$7 - 5 = 2$$, and $$9 - 7 = 2$$. - The difference between consecutive $$y$$-values is also constant: $$33 - 11 = 22$$, $$66 - 33 = 33$$, and $$88 - 66 = 22$$. Since the differences in $$y$$-values are not constant, **Table A does not represent a linear function**. --- ### **Table B:** $$\begin{array}{|c|c|} \hline x & y \\ \hline 2 & 44 \\ 3 & 22 \\ 5 & 0 \\ 7 & -2 \\ \hline \end{array}$$ - The difference between $$x$$-values: $$3 - 2 = 1$$, $$5 - 3 = 2$$, $$7 - 5 = 2$$. - The difference between consecutive $$y$$-values: $$22 - 44 = -22$$, $$0 - 22 = -22$$, $$-2 - 0 = -2$$. Since the differences between the $$y$$-values are not constant, **Table B does not represent a linear function**. --- ### **Table C:** $$\begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ 2 & 33 \\ 4 & 66 \\ 6 & 88 \\ \hline \end{array}$$ - The difference between $$x$$-values is constant: $$2 - 0 = 2$$, $$4 - 2 = 2$$, and $$6 - 4 = 2$$. - The difference between consecutive $$y$$-values is also constant: $$33 - 0 = 33$$, $$66 - 33 = 33$$, and $$88 - 66 = 22$$. Since the differences in $$y$$-values are not constant, **Table C does not represent a linear function**. --- ### **Table D:** $$\begin{array}{|c|c|} \hline x & y \\ \hline 0 & 44 \\ 2 & 33 \\ 4 & 22 \\ 6 & 11 \\ \hline \end{array}$$ - The difference between $$x$$-values is constant: $$2 - 0 = 2$$, $$4 - 2 = 2$$, and $$6 - 4 = 2$$. - The difference between consecutive $$y$$-values is also constant: $$33 - 44 = -11$$, $$22 - 33 = -11$$, and $$11 - 22 = -11$$. Since the difference in $$y$$-values is constant, **Table D represents a linear function**. --- ### Conclusion: **Table D** represents a linear function because both the $$x$$-values and $$y$$-values change in a consistent, linear manner. --- Would you like a deeper explanation of linear functions, or do you have any further questions? Here are some related questions for you: 1. How can you find the slope of a linear function from a table of values? 2. What does the slope of a linear function tell us about the relationship between $$x$$ and $$y$$? 3. How can you write the equation of a linear function from a table? 4. What is the significance of the y-intercept in a linear equation? 5. Can a linear function have a slope of zero, and what would that look like? **Tip**: When checking if a table represents a linear function, always look for constant differences in both $$x$$-values and $$y$$-values!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Slope
Table of Values
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Theorems
Linear Function Definition
Suitable Grade Level
Grades 7-9