To find the equation of the linear function in slope-intercept form $y = mx + b$, we need to determine two key components:

1. Slope $m$ (rate of change): The slope can be calculated by taking any two points $(x_1, y_1)$ and $(x_2, y_2)$ from the table and using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

2. Intercept $b$ (y-intercept): Once we know the slope, we can substitute a point $(x_1, y_1)$ into the slope-intercept equation to solve for $b$.

Step 1: Calculate the slope $m$

Using the points $(-1, 6)$ and $(3, -6)$ from the table: $m = \frac{-6 - 6}{3 - (-1)} = \frac{-12}{4} = -3$

Step 2: Find the y-intercept $b$

Now that we know $m = -3$, use one of the points to find $b$. Let's use the point $(-1, 6)$: $y = mx + b \implies 6 = -3(-1) + b \implies 6 = 3 + b$ $b = 6 - 3 = 3$

Final Equation

The equation of the linear function is: $y = -3x + 3$

Would you like more details on this process or explanations of any specific steps?

Here are 5 related questions for further practice:

1. How do you find the slope if the table has more points?
2. What is the meaning of a negative slope in a linear equation?
3. How do you interpret the y-intercept graphically?
4. Can you find the equation using other pairs of points from the table?
5. How would you graph the equation $y = -3x + 3$?

Tip: The slope tells you how much $y$ changes for a given change in $x$, and it can be understood as the "steepness" of the line.