To find the linear function that passes through the points $(1, -1)$ and $(3, 3)$, we need to determine the slope and then use the point-slope form of a linear equation.

Step 1: Find the slope $m$

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Using the points $(1, -1)$ and $(3, 3)$: $m = \frac{3 - (-1)}{3 - 1} = \frac{3 + 1}{2} = \frac{4}{2} = 2$ So, the slope $m = 2$.

Step 2: Use the point-slope form of a line

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Using the slope $m = 2$ and the point $(1, -1)$: $y - (-1) = 2(x - 1)$ Simplifying: $y + 1 = 2(x - 1)$ $y + 1 = 2x - 2$ Subtract 1 from both sides: $y = 2x - 3$

Conclusion:

The linear function is: $y = 2x - 3$

Would you like more details or further clarification?

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Here are 5 related questions to expand your understanding:

1. How do you find the slope if the line is vertical?
2. How would you find the equation if given three points?
3. Can you use this method for a nonlinear function, and if not, why?
4. What happens if the two points are the same?
5. How can you rewrite the equation in standard form?

Tip: Always double-check your calculations by substituting both points into the final equation.