To fill in the missing value in the table, we can analyze the graph and find the function's behavior.

Step 1: Identify the Slope

The line on the graph passes through the points:

• $(2, 5)$
• $(8, 1)$

Using the formula for slope $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1) = (2, 5)$ and $(x_2, y_2) = (8, 1)$: $m = \frac{1 - 5}{8 - 2} = \frac{-4}{6} = -\frac{2}{3}$

Step 2: Write the Equation of the Line

Since the slope $m$ is $-\frac{2}{3}$, we can use the point-slope form $y - y_1 = m(x - x_1)$ with point $(2, 5)$ to get: $y - 5 = -\frac{2}{3}(x - 2)$ Expanding this equation: $y - 5 = -\frac{2}{3}x + \frac{4}{3}$ $y = -\frac{2}{3}x + \frac{4}{3} + 5$ Converting 5 to a fraction $\frac{15}{3}$, we get: $y = -\frac{2}{3}x + \frac{19}{3}$

Step 3: Find the Missing Output for $x = 5$

Now, substitute $x = 5$ into the equation: $y = -\frac{2}{3}(5) + \frac{19}{3}$ $y = -\frac{10}{3} + \frac{19}{3} = \frac{9}{3} = 3$

Final Answer

The missing output value when the input is $5$ is $3$.

Completed Table:

Would you like further details on this process, or do you have any other questions?

1. What is the importance of the slope in finding the equation of a line?
2. How does the point-slope form of a line differ from the slope-intercept form?
3. Can you use two points to find the equation of any line?
4. Why is the function here represented by a straight line?
5. How does changing the slope affect the line on the graph?

Tip: When interpreting graphs, identifying two points and calculating the slope is an effective way to derive the function’s equation.