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

Step 1: Find the Slope $m$

The slope $m$ can be calculated using two points $(x_1, y_1)$ and $(x_2, y_2)$ with the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's choose two points from the table:

• Point 1: $(-3, 6)$
• Point 2: $(2, -9)$

Plugging in the values: $m = \frac{-9 - 6}{2 - (-3)} = \frac{-15}{5} = -3$

So, the slope $m$ is $-3$.

Step 2: Find the Y-Intercept $b$

Using the slope-intercept form $y = mx + b$, substitute $m = -3$ and use one of the points to solve for $b$.

Using the point $(2, -9)$: $-9 = (-3)(2) + b$ $-9 = -6 + b$ $b = -9 + 6 = -3$

Step 3: Write the Equation

Now we have both the slope and y-intercept: $y = -3x - 3$

Final Answer

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

Would you like more details on any specific step, or have any questions?

Here are some related questions for further exploration:

1. How would the slope calculation change if we used different points from the table?
2. Can you explain why this function is linear based on the table data?
3. What would the graph of this equation look like?
4. How does the y-intercept affect the position of the line on the graph?
5. How would the equation change if the slope was positive?

Tip: In linear functions, consistency in the slope across any two points confirms that the function is indeed linear.