To write the equation of the function, we can use the point-slope form of a linear equation. The point-slope form is given by:

$y - y_1 = m(x - x_1)$

where:

• $m$ is the slope (rate of change),
• $(x_1, y_1)$ is a point on the line.

Given:

• Rate of change (slope) $m = \frac{1}{2}$,
• Point $(x_1, y_1) = (2, 4)$.

Substituting these values into the point-slope form:

$y - 4 = \frac{1}{2}(x - 2)$

Now, simplify the equation:

$y - 4 = \frac{1}{2}x - 1$

Adding 4 to both sides to solve for $y$:

$y = \frac{1}{2}x + 3$

So, the equation of the function is:

$y = \frac{1}{2}x + 3$

Would you like more details or have any questions? Here are some related questions to explore:

1. How would the equation change if the slope were $\frac{3}{4}$ instead of $\frac{1}{2}$?
2. What is the y-intercept of the function $y = \frac{1}{2}x + 3$?
3. How can you find the x-intercept of the function $y = \frac{1}{2}x + 3$?
4. What does the slope of $\frac{1}{2}$ tell you about the function's graph?
5. How do you convert the point-slope form to the slope-intercept form?

Tip: Always check if the equation of the line correctly satisfies the given point by substituting the values of $x$ and $y$ back into the final equation.