To solve the problem, we need to find the equation of a linear function $f(x)$ with:

• Slope $m = -\frac{5}{7}$,
• $y$-intercept $(0, \frac{1}{3})$.

The general formula for a linear function is:

$f(x) = mx + b$

Here:

• $m = -\frac{5}{7}$ (slope),
• $b = \frac{1}{3}$ (y-intercept).

Substituting the values:

$f(x) = -\frac{5}{7}x + \frac{1}{3}$

Thus, the formula for the function is: $f(x) = -\frac{5}{7}x + \frac{1}{3}$

Would you like further explanation on this?

Here are 5 related questions to expand on this topic:

1. What is the general formula for finding the equation of a line given two points?
2. How can we convert this linear function to standard form $Ax + By = C$?
3. How do changes in the slope $m$ affect the graph of a line?
4. What does the $y$-intercept represent graphically?
5. How can we calculate the $x$-intercept of this line?

Tip: Always remember that the slope represents the "rise over run," indicating how the line moves vertically versus horizontally!