To graph the function $f(x)$ and the secant lines passing through the point $P(4, 2)$ and the points $Q(x, f(x))$ for $x$-values of 3, 6, and 8, we need the specific function $f(x)$. However, I can walk you through the general steps and formulas used to find the slopes of these secant lines.

General Steps

1. Identify the points on the graph:

   • $P(4, 2)$: This is the fixed point through which all secant lines will pass.
   • $Q(x, f(x))$ for $x = 3$, $x = 6$, and $x = 8$: These are the points through which the secant lines will pass.

2. Calculate the slope of each secant line: The slope $m$ of the secant line passing through $P(4, 2)$ and $Q(x, f(x))$ is given by the difference quotient: $m = \frac{f(x) - 2}{x - 4}$ where $f(x)$ is the value of the function at the given $x$-value.

Calculation of Slopes (Example with Hypothetical Function $f(x)$)

If $f(x)$ is known, substitute the values to find the slopes. Suppose $f(x) = x^2 - 3x + 5$. Then:

• For $x = 3$: $f(3) = 3^2 - 3(3) + 5 = 9 - 9 + 5 = 5$ The slope is: $m = \frac{5 - 2}{3 - 4} = \frac{3}{-1} = -3.000$

• For $x = 6$: $f(6) = 6^2 - 3(6) + 5 = 36 - 18 + 5 = 23$ The slope is: [ m = \frac{23 - 2}{6 - 4} = \frac{21}{2}