Given the function $f(x) = 5x - 7$, we are tasked with finding:

(a) The average rate of change from $x = 5$ to $x = 8$

The formula for the average rate of change of a function between two points $x = a$ and $x = b$ is:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Here, $a = 5$ and $b = 8$. First, we need to calculate $f(5)$ and $f(8)$:

• $f(5) = 5(5) - 7 = 25 - 7 = 18$
• $f(8) = 5(8) - 7 = 40 - 7 = 33$

Now, using the formula:

$\text{Average Rate of Change} = \frac{f(8) - f(5)}{8 - 5} = \frac{33 - 18}{3} = \frac{15}{3} = 5$

So, the average rate of change is $5$.

(b) Equation of the secant line through $(5, f(5))$ and $(8, f(8))$

The secant line is a straight line passing through the two points $(5, f(5))$ and $(8, f(8))$. These points are:

• $(5, f(5)) = (5, 18)$
• $(8, f(8)) = (8, 33)$

To find the equation of the secant line, we use the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

Where $m$ is the slope (which is the average rate of change calculated earlier), and $(x_1, y_1)$ is one of the points on the line (we can use either $(5, 18)$ or $(8, 33)$).

We already know that the slope $m = 5$, so using the point $(5, 18)$:

$y - 18 = 5(x - 5)$

Simplifying:

$y - 18 = 5x - 25$ $y = 5x - 25 + 18$ $y = 5x - 7$

Thus, the equation of the secant line is $y = 5x - 7$.

Summary:

• (a) The average rate of change is $5$.
• (b) The equation of the secant line is $y = 5x - 7$.

Would you like more details or have any questions?

Here are some related questions:

1. What is the significance of the average rate of change for linear functions?
2. How does the secant line differ from a tangent line in calculus?
3. What happens if we compute the average rate of change over a larger interval for nonlinear functions?
4. How would the process change if the function was quadratic?
5. What is the geometric interpretation of the average rate of change?

Tip: For linear functions, the secant line between any two points will always be the same as the line itself!