Math Problem Statement
Which table of values represents a linear function?
A
xx
yy
00
minus, 6−6
11
minus, 3−3
22
minus, 1−1
33
11
B
xx
yy
minus, 4−4
11
22
44
44
66
66
77
C
xx
yy
11
77
22
11
33
minus, 4−4
44
minus, 9−9
D
xx
yy
minus, 1−1
77
33
55
55
44
99
22
Answer
Solution
To determine which table of values represents a linear function, we need to check the consistency of the slope between each pair of points. For a function to be linear, the difference in values divided by the difference in values (the slope) should be constant.
Let's analyze each table:
Table A:
x & y \\ \hline 0 & -6 \\ 1 & -3 \\ 2 & -1 \\ 3 & 1 \\ \end{array}$$ **Changes:** $$\text{slope between } (0, -6) \text{ and } (1, -3): \frac{-3 - (-6)}{1 - 0} = \frac{3}{1} = 3$$ $$\text{slope between } (1, -3) \text{ and } (2, -1): \frac{-1 - (-3)}{2 - 1} = \frac{2}{1} = 2$$ $$\text{slope between } (2, -1) \text{ and } (3, 1): \frac{1 - (-1)}{3 - 2} = \frac{2}{1} = 2$$ The slopes are not consistent, so **Table A is not linear**. --- **Table B:** $$\begin{array}{c|c} x & y \\ \hline -4 & 1 \\ 2 & 4 \\ 4 & 6 \\ 6 & 7 \\ \end{array}$$ **Changes:** $$\text{slope between } (-4, 1) \text{ and } (2, 4): \frac{4 - 1}{2 - (-4)} = \frac{3}{6} = \frac{1}{2}$$ $$\text{slope between } (2, 4) \text{ and } (4, 6): \frac{6 - 4}{4 - 2} = \frac{2}{2} = 1$$ $$\text{slope between } (4, 6) \text{ and } (6, 7): \frac{7 - 6}{6 - 4} = \frac{1}{2}$$ The slopes are not consistent, so **Table B is not linear**. --- **Table C:** $$\begin{array}{c|c} x & y \\ \hline 1 & 7 \\ 2 & 1 \\ 3 & -4 \\ 4 & -9 \\ \end{array}$$ **Changes:** $$\text{slope between } (1, 7) \text{ and } (2, 1): \frac{1 - 7}{2 - 1} = \frac{-6}{1} = -6$$ $$\text{slope between } (2, 1) \text{ and } (3, -4): \frac{-4 - 1}{3 - 2} = \frac{-5}{1} = -5$$ $$\text{slope between } (3, -4) \text{ and } (4, -9): \frac{-9 - (-4)}{4 - 3} = \frac{-5}{1} = -5$$ The slopes are not consistent, so **Table C is not linear**. --- **Table D:** $$\begin{array}{c|c} x & y \\ \hline -1 & 7 \\ 3 & 5 \\ 5 & 4 \\ 9 & 2 \\ \end{array}$$ **Changes:** $$\text{slope between } (-1, 7) \text{ and } (3, 5): \frac{5 - 7}{3 - (-1)} = \frac{-2}{4} = -\frac{1}{2}$$ $$\text{slope between } (3, 5) \text{ and } (5, 4): \frac{4 - 5}{5 - 3} = \frac{-1}{2} = -\frac{1}{2}$$ $$\text{slope between } (5, 4) \text{ and } (9, 2): \frac{2 - 4}{9 - 5} = \frac{-2}{4} = -\frac{1}{2}$$ The slopes are consistent, so **Table D is linear**. --- **Conclusion:** **Table D** represents a linear function. Would you like more details or have any questions? --- **Related Questions:** 1. How do you graph a linear function from a table of values? 2. What is the significance of the slope in a linear function? 3. How can you tell if a function is linear just by looking at its equation? 4. Can a linear function have a slope of zero? 5. How does the slope relate to the steepness of a line? **Tip:** The slope of a linear function is the same between any two points on the function's graph.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Slope
Formulas
Slope formula: \( m = \frac{y2 - y1}{x2 - x1} \)
Theorems
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Suitable Grade Level
Grades 7-9