To find the equation of a linear function $y = mx + b$ (slope-intercept form), we need to determine:

1. The slope $m$.
2. The $y$-intercept $b$.

Step 1: Calculate the slope $m$

The slope $m$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}.$ Using two points from the table, say $(-1, 3)$ and $(2, 9)$: $m = \frac{9 - 3}{2 - (-1)} = \frac{6}{3} = 2.$

Step 2: Solve for $b$

Now that $m = 2$, substitute a point from the table (e.g., $(-1, 3)$) into the equation $y = mx + b$ to solve for $b$: $3 = 2(-1) + b \implies 3 = -2 + b \implies b = 5.$

Final Equation

The equation of the linear function is: $y = 2x + 5.$

Verification

Check the equation with other points in the table:

• For $x = 2$, $y = 2(2) + 5 = 9$ ✓
• For $x = 5$, $y = 2(5) + 5 = 15$ ✓
• For $x = 8$, $y = 2(8) + 5 = 21$ ✓

The equation $y = 2x + 5$ works for all points.

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Would you like further details or explanations?

Related Questions:

1. How can the slope formula be derived geometrically?
2. What are some examples of real-world applications of linear functions?
3. How would the equation change if the $y$-intercept were negative?
4. Can the equation $y = 2x + 5$ be represented in point-slope form?
5. How do you identify a linear relationship from a table without calculating?

Tip: Always verify the slope calculation using more than two points to ensure consistency!