To determine which tables of values represent linear functions, we need to check whether the differences in $y$-values are consistent for equal differences in $x$-values. A linear function will have a constant rate of change, meaning the difference between consecutive $y$-values divided by the difference between consecutive $x$-values will always be the same.

Let's check each option:

Option 1:

x: &\quad -1, 0, 1, 2 \\ y: &\quad -2.5, -3, -3.5, -4 \\ \end{aligned}$$ - Change in $$x$$: $$1 - (-1) = 1$$ - Change in $$y$$: $$-3 - (-2.5) = -0.5$$, $$-3.5 - (-3) = -0.5$$, $$-4 - (-3.5) = -0.5$$ Since the change in $$y$$ is constant, this table represents a **linear function**. ### Option 2: $$\begin{aligned} x: &\quad -1, 0, 1, 2 \\ y: &\quad 3, 5, 7, 9 \\ \end{aligned}$$ - Change in $$x$$: $$1 - (-1) = 1$$ - Change in $$y$$: $$5 - 3 = 2$$, $$7 - 5 = 2$$, $$9 - 7 = 2$$ The change in $$y$$ is constant, so this also represents a **linear function**. ### Option 3: $$\begin{aligned} x: &\quad -1, 0, 1, 2 \\ y: &\quad 2, -2, 2, -2 \\ \end{aligned}$$ The $$y$$-values alternate between 2 and -2, which means the differences are not constant, so this is **not a linear function**. ### Option 4: $$\begin{aligned} x: &\quad -2, 0, 2, 4 \\ y: &\quad -5, 0, 10, 20 \\ \end{aligned}$$ - Change in $$x$$: $$2 - (-2) = 4$$ - Change in $$y$$: $$0 - (-5) = 5$$, $$10 - 0 = 10$$, $$20 - 10 = 10$$ The rate of change is not constant, so this is **not a linear function**. ### Option 5: $$\begin{aligned} x: &\quad -1, 0, 1, 2 \\ y: &\quad -0.5, 0, -0.5, -2 \\ \end{aligned}$$ The differences in $$y$$-values are inconsistent, so this is **not a linear function**. ### Conclusion: The tables representing linear functions are **Option 1** and **Option 2**. Would you like more details or have any questions? Here are some follow-up questions: 1. How can you determine a linear function from an equation? 2. What are the characteristics of a non-linear function? 3. How do you find the slope from a table of values? 4. Can a linear function have a negative slope? 5. What real-life situations can be modeled by linear functions? **Tip:** To quickly identify a linear function, look for a constant rate of change in the $$y$$-values relative to the $$x$$-values.