To find the equation of the linear function represented by the table, let's analyze the values provided.

The points given in the table are: $(-1, 3), (1, 5), (3, 7), (5, 9)$

The general form for a linear equation is: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

Step 1: Calculate the Slope ( $m$ )

The slope $m$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(1, 5)$ and $(3, 7)$: $m = \frac{7 - 5}{3 - 1} = \frac{2}{2} = 1$

So, the slope $m$ is $1$.

Step 2: Determine the y-intercept ( $b$ )

Now that we know $m = 1$, we can use one of the points to find $b$.

Using the point $(1, 5)$: $y = mx + b$ $5 = 1 \cdot 1 + b$ $5 = 1 + b$ $b = 4$

Final Equation

With $m = 1$ and $b = 4$, the equation of the linear function is: $y = x + 4$

Would you like more details, or do you have any questions?

Here are some related questions for further exploration:

1. How do you verify that this equation is correct for all points in the table?
2. How would you find the equation if the points were non-linear?
3. What are other forms of a linear equation besides slope-intercept form?
4. How would the equation change if the slope was negative?
5. How can you graph this equation based on the slope and intercept?

Tip: To quickly find the y-intercept when you know the slope, substitute a known point into the equation $y = mx + b$ and solve for $b$.