To find the equation of a linear function from a table of values, we start by calculating the slope ($m$) and then determine the y-intercept ($b$) to express it in slope-intercept form, $y = mx + b$.

Step 1: Calculate the Slope (m)

Using two points from the table, we can calculate the slope with the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Let’s pick the points $(1, 2)$ and $(2, 7)$: $m = \frac{7 - 2}{2 - 1} = \frac{5}{1} = 5$

So, the slope $m = 5$.

Step 2: Find the y-intercept (b)

Now that we know $m = 5$, we can substitute one of the points into the equation $y = mx + b$ to find $b$.

Using the point $(1, 2)$: $2 = 5(1) + b$ $2 = 5 + b$ $b = 2 - 5 = -3$

Step 3: Write the Equation

Now we have both the slope $m = 5$ and the y-intercept $b = -3$. Therefore, the equation of the line is: $y = 5x - 3$

Final Answer

The equation of the linear function is: $y = 5x - 3$

Would you like a further breakdown, or do you have any questions?

Here are five related questions that build on this:

1. How do you calculate the slope between two points if they are given as fractions?
2. How do you identify whether a table of values represents a linear function?
3. What would the equation look like if the slope was negative?
4. How can you determine the equation if only given one point and the slope?
5. How would the equation change if the values in the table represented a different constant slope?

Tip: When determining the slope, it's helpful to check between multiple points to ensure consistency in a linear relationship.