(a)

Graph *f* and the secant lines passing through the point *P*(4, 2) and *Q*(*x*, *f*(*x*)) for *x*-values of 3, 6, and 8.

Find the slope of each secant line. (Round your answers to three decimal places.)

line passing through 

*Q*(3, *f*(*x*))

line passing through 

*Q*(6, *f*(*x*))

line passing through 

*Q*(8, *f*(*x*))

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}

Learn how to calculate the slopes of secant lines passing through the fixed point P(4, 2) and points Q(x, f(x)) for x-values 3, 6, and 8. Detailed step-by-step solution with examples provided.

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