To find the linear function in standard form that passes through the points (4, 4) and (6, 5), we first determine the equation of the line using these points.

Step 1: Find the slope of the line

The slope formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the points $(4, 4)$ and $(6, 5)$:

$m = \frac{5 - 4}{6 - 4} = \frac{1}{2}$

So, the slope of the line is $\frac{1}{2}$.

Step 2: Use the point-slope form of the equation

The point-slope form of a linear equation is:

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

Using the point $(4, 4)$ and the slope $m = \frac{1}{2}$:

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

Step 3: Simplify the equation

First, distribute on the right side:

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

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

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

Step 4: Convert to standard form

To write the equation in standard form $Ax + By = C$, eliminate the fraction by multiplying the entire equation by 2:

$2y = x + 4$

Now, rearrange it into standard form:

$x - 2y = -4$

This matches option D.

Final Answer:

The correct answer is D. $x - 2y = -4$.

Would you like further details or have any other questions?

Here are some related questions for further exploration:

1. How do you find the slope given two points?
2. What is the general process of converting slope-intercept form to standard form?
3. How can you determine if a line is parallel to another using slope?
4. Can you find the equation of a line passing through two points in intercept form?
5. How do we interpret the coefficients of $x$ and $y$ in the standard form?

Tip: When converting from slope-intercept form to standard form, always clear any fractions first.