To find the equation of the linear function represented by the table, we need to determine the slope ($m$) and the y-intercept ($b$) of the equation in slope-intercept form, $y = mx + b$.

Step 1: Find the Slope $m$

The slope is calculated by the change in $y$ values divided by the change in $x$ values.

Looking at the table:

• From $x = 1$ to $x = 2$, $y$ changes from 0 to -5. $m = \frac{-5 - 0}{2 - 1} = \frac{-5}{1} = -5$

So, the slope $m$ is $-5$.

Step 2: Find the Y-Intercept $b$

Using the point $(1, 0)$: $y = mx + b$ $0 = (-5)(1) + b$ $0 = -5 + b$ $b = 5$

Final Equation

With $m = -5$ and $b = 5$, the equation of the line is: $y = -5x + 5$

Answer:

The equation of the linear function is: $y = -5x + 5$

Would you like further details on any of these steps?

Here are some related questions for further exploration:

1. How would you find the equation if only two points were given?
2. What does the slope represent in this context?
3. How would you graph this equation based on the table values?
4. How would the equation change if the values were positive?
5. How do you verify if a function is linear from a table of values?

Tip: When finding the slope from a table, always check if the change in $y$ values over $x$ values is consistent, confirming it’s a linear function.