\( 1 \leftarrow \) Question \( 9,3.5 .29 \) Find an equation of the line containing the given points. Use function notation to write the equation. \( \left(\frac{5}{7}, \frac{3}{7}\right) \) and \( \left(-\frac{1}{7}, \frac{9}{14}\right) \) \( f(x)=\square \) \( ( \) Simplify your answer. Use integers or fractions for any numbers in the 0 of 1

To find the equation of the line containing the given points \(\left(\frac{5}{7}, \frac{3}{7}\right)\) and \(\left(-\frac{1}{7}, \frac{9}{14}\right)\), we first need to determine the slope \(m\). The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the values: \[ m = \frac{\frac{9}{14} - \frac{3}{7}}{-\frac{1}{7} - \frac{5}{7}} = \frac{\frac{9}{14} - \frac{6}{14}}{-\frac{6}{7}} = \frac{\frac{3}{14}}{-\frac{6}{7}} = -\frac{3}{12} = -\frac{1}{4} \] Now, using point-slope form \(y - y_1 = m(x - x_1)\) with one of the points, say \(\left(\frac{5}{7}, \frac{3}{7}\right)\): \[ y - \frac{3}{7} = -\frac{1}{4}\left(x - \frac{5}{7}\right) \] To express \(f(x)\): \[ f(x) = -\frac{1}{4}x + \frac{5}{28} + \frac{3}{7} \] Converting \(\frac{3}{7}\) to \(\frac{12}{28}\), we get: \[ f(x) = -\frac{1}{4}x + \frac{17}{28} \] So, the function notation becomes: \[ f(x) = -\frac{1}{4}x + \frac{17}{28} \] --- Understanding slope can greatly enhance your ability to interpret and graph linear equations. The slope indicates how steep a line is; if positive, the line rises from left to right, and if negative, it falls. This concept not only applies to mathematics but also is crucial in various fields like economics and physics. When working with coordinates, it’s easy to make mistakes, especially with negative signs and fractions. One common error is miscalculating the change in \(y\) or \(x\). Always double-check your addition and subtraction, especially with fractions. Use visual aids to sketch the points before continuing with calculations—this can help clarify relationships and prevent errors.