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 $b$ from the table of values.

Step 1: Find the Slope ($m$)

The slope $m$ of a linear function is calculated by using the formula:

$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$

Let's take two points from the table to calculate the slope:

• Point 1: $(1, 0)$
• Point 2: $(2, -4)$

Substitute these values into the formula:

$m = \frac{{-4 - 0}}{{2 - 1}} = \frac{{-4}}{{1}} = -4$

So, the slope $m$ is -4.

Step 2: Find the y-Intercept ($b$)

In slope-intercept form $y = mx + b$, we can find $b$ by substituting one of the points and the slope into the equation.

Using the point $(1, 0)$: $0 = -4 \cdot 1 + b$ $0 = -4 + b$ $b = 4$

Final Equation

The equation of the linear function is:

$y = -4x + 4$

Let me know if you want further details or have any questions!

Here are some additional questions to deepen your understanding:

1. How would you find the equation if the points in the table did not form a linear function?
2. What does the slope represent in a real-world context if this function represented a real scenario?
3. How would you determine if a function is increasing or decreasing based on its slope?
4. If the y-intercept were different, how would that affect the graph of the equation?
5. Can you use the same steps to find the equation of a line if only given two points without a table?

Tip: Always use different pairs of points to verify the slope in a linear function; it should be consistent for any two points on the line.