Math Problem Statement
Which of the following tables represents a linear function?
x 1 1 1 1 1 y −3 −2 −1 0 1
x −4 −2 0 2 4 y 4 2 0 2 4
x −5 −3 −1 1 3 y negative one half 1 2 7 over 2 5
x −6 −4 −2 0 2 y 5 13 over 3 11 over 3 3 7 over 3
Solution
To determine which table represents a linear function, we need to check if the relationship between and can be expressed in the form , where is the slope and is the y-intercept. This requires that the differences between consecutive -values are constant when the differences between consecutive -values are constant.
Let's analyze each table:
Table 1
x & 1 & 1 & 1 & 1 & 1 \\ \hline y & -3 & -2 & -1 & 0 & 1 \\ \end{array}$$ In this table, $$x$$ is constant, so this cannot be a function. This table does not represent a linear function. ### Table 2 $$\begin{array}{c|ccccc} x & -4 & -2 & 0 & 2 & 4 \\ \hline y & 4 & 2 & 0 & 2 & 4 \\ \end{array}$$ Calculate the differences: - $$\Delta y = 2 - 4 = -2$$ - $$\Delta y = 0 - 2 = -2$$ - $$\Delta y = 2 - 0 = 2$$ - $$\Delta y = 4 - 2 = 2$$ The differences between $$y$$-values are not consistent (they alternate between -2 and 2), so this table does not represent a linear function. ### Table 3 $$\begin{array}{c|ccccc} x & -5 & -3 & -1 & 1 & 3 \\ \hline y & -\frac{1}{2} & 1 & 2 & \frac{7}{2} & 5 \\ \end{array}$$ Calculate the differences: - $$\Delta y = 1 - \left(-\frac{1}{2}\right) = 1 + \frac{1}{2} = \frac{3}{2}$$ - $$\Delta y = 2 - 1 = 1$$ - $$\Delta y = \frac{7}{2} - 2 = \frac{7}{2} - \frac{4}{2} = \frac{3}{2}$$ - $$\Delta y = 5 - \frac{7}{2} = \frac{10}{2} - \frac{7}{2} = \frac{3}{2}$$ The differences are $$\frac{3}{2}$$, 1, $$\frac{3}{2}$$, $$\frac{3}{2}$$, so this table does not represent a linear function as the differences between $$y$$-values are not consistent. ### Table 4 $$\begin{array}{c|ccccc} x & -6 & -4 & -2 & 0 & 2 \\ \hline y & 5 & \frac{13}{3} & \frac{11}{3} & 3 & \frac{7}{3} \\ \end{array}$$ Calculate the differences: - $$\Delta y = \frac{13}{3} - 5 = \frac{13}{3} - \frac{15}{3} = -\frac{2}{3}$$ - $$\Delta y = \frac{11}{3} - \frac{13}{3} = -\frac{2}{3}$$ - $$\Delta y = 3 - \frac{11}{3} = \frac{9}{3} - \frac{11}{3} = -\frac{2}{3}$$ - $$\Delta y = \frac{7}{3} - 3 = \frac{7}{3} - \frac{9}{3} = -\frac{2}{3}$$ The differences between $$y$$-values are consistent, all being $$-\frac{2}{3}$$. Therefore, this table represents a linear function with a slope of $$-\frac{2}{3}$$. So, the table that represents a linear function is: $$\begin{array}{c|ccccc} x & -6 & -4 & -2 & 0 & 2 \\ \hline y & 5 & \frac{13}{3} & \frac{11}{3} & 3 & \frac{7}{3} \\ \end{array}$$ Would you like more details or have any questions? Here are 5 questions you might want to ask next: 1. How do you determine the slope of a linear function from a table? 2. Can you explain how to convert a sequence of points into the equation of a linear function? 3. How do you identify a non-linear function from a table of values? 4. What are some real-world examples of linear functions? 5. How can you graph a linear function given a table of values? **Tip:** To check if a function is linear, ensure the differences between the $$y$$-values are consistent when the differences between the $$x$$-values are consistent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear functions
Slope
Consistency of differences
Formulas
-
Theorems
-
Suitable Grade Level
Grades 7-9