Let's find the equation of the line that passes through the points $(5.7, -3)$ and $(8.7, 9.4)$ in slope-intercept form, $y = mx + b$.

Step 1: Find the slope (m)

The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

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

Substitute the given points $(5.7, -3)$ and $(8.7, 9.4)$:

$m = \frac{9.4 - (-3)}{8.7 - 5.7} = \frac{9.4 + 3}{8.7 - 5.7} = \frac{12.4}{3} = 4.13$

Step 2: Use the slope-intercept form

We now use the slope-intercept form $y = mx + b$. To find the y-intercept $b$, substitute one of the points and the slope into the equation. Let's use the point $(5.7, -3)$:

$y = mx + b$

$-3 = 4.13(5.7) + b$

Now, solve for $b$:

$-3 = 23.541 + b$

$b = -3 - 23.541 = -26.541$

So the equation of the line is approximately:

$y = 4.13x - 26.54$

Final Answer:

The closest option to this result is:

$\boxed{D. \ f(x) = 4.13x - 26.560}$

Would you like more details or have any questions?

Here are 5 related questions you might find useful:

1. How can I derive the slope formula from the equation of a line?
2. What is the difference between slope-intercept form and point-slope form?
3. How do you graph a line given its equation in slope-intercept form?
4. What happens if two points have the same x-value or y-value when calculating the slope?
5. How does the slope affect the steepness and direction of a line?

Tip: Always round your intermediate values carefully to avoid significant rounding errors in the final result!