To find the equation of the linear function, we need to determine both the slope ($m$) and the y-intercept ($b$) of the line. Here’s how we can solve it:

Step 1: Determine the Slope ($m$)

The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(0, 4)$ and $(1, 6)$ from the table: $m = \frac{6 - 4}{1 - 0} = \frac{2}{1} = 2$

So, the slope $m = 2$.

Step 2: Identify the Y-intercept ($b$)

The y-intercept $b$ is the value of $y$ when $x = 0$. From the table, we see that when $x = 0$, $y = 4$. Therefore, the y-intercept $b = 4$.

Step 3: Write the Equation of the Line

The equation of a linear function in slope-intercept form is: $y = mx + b$ Substituting the values of $m = 2$ and $b = 4$: $y = 2x + 4$

Answer Summary

• Slope: $2$
• Y-intercept: $4$
• Equation: $y = 2x + 4$

Would you like more details or have any questions?

Here are five related questions that could help deepen your understanding:

1. How would the equation change if the slope were different?
2. How can we determine the slope if only one point and the y-intercept are given?
3. How do we know if a function is linear from just a table of values?
4. What happens to the y-intercept if all values of $y$ are increased by a constant?
5. How can we verify that all points in the table lie on the line $y = 2x + 4$?

Tip: Always check multiple pairs of points to confirm the slope is consistent across a table for a linear function.